Motion

Abstract

The study of human motion begins with classifying the many types of locomotion and continues with descriptions of muscles in the body. The analysis of standing starts with a discussion of the overall and local stability of the body and of friction, and continues with models of walking. This includes a review of harmonic motion and pendulums. These concepts are applied to running, which is also compared to walking. The energetics and dynamics of jumping and the pole vault are followed by modeling how to throw balls, such as baseballs, and swing objects, such as golf clubs. This is then related to a more formal treatment of multisegment modeling. Collisions of the body due to falls, in contact sports such as boxing and karate, and in hitting balls are investigated. The flight of balls that are thrown or hit is analyzed, including how it is affected by drag and ball spinning. The effect of friction is probed in skiing and in the bouncing and rolling of balls.

Problems

Stability, Friction, Human Moments of Inertia, and Muscles

3.1

Pregnancy StabilityConsider a person modeled as a one-dimensional being of height H and mass $m_{\mathrm {b}}$ in the two-dimensional world of Fig. 3.8a. The person has a constant mass per unit length and the body rests 25% from the back of her massless feet, which have length $0.152\,H$ (as from Table 1.6) and negligible height. To what angles can the person tilt in the forward and backward directions and still maintain overall balance and stability? (Consider potential rotations about the front and back of the feet, as in Fig. 3.8a.)

3.2

Pregnancy StabilityCan the pregnant woman, modeled in the two-dimensional rigid-figure world of Fig. 3.8b, maintain overall balance and stability? Assume the centers of mass of the woman without the womb and the womb itself are both $0.58\,H$ above her feet. The additional mass is $0.25\,m_{\mathrm {b}}$ and has a center of mass $0.1\,H$ in front of the rest of her body. (Assume the other information given about the feet in Problem 3.1.)

3.3

Gymnastics StabilityRefer to the gymnast in the arabesque position described in Problem 2.11. In attaining this position her center of mass descends but does not move laterally. How does this affect her overall stability?

3.4

StabilityA person is supported by two crutches. Sketch diagrams of the patterns of the person’s feet and the bottoms of the two crutches, when the cructches are held either in front of, behind, or to either side of the person. Describe the stability provided by each arrangement.

3.5

StabilityWhen you stand on a moving bus, how do you arrange your feet to try to maintain stability and why?

3.6

Football StabilityIn football, lineman often line up near the line of scrimmage before the play starts crouching in a three-point stance (two feet and one hand on the ground) or a four-point stance (two feet and two hands on the ground), and when the play starts theyPulling, pushing push forward with their legs [60].

(a) Explain why these stances help them push forward well and maintain a low center of mass.

(b) For each stance, sketch a diagram of the ground showing the areas spanned by their feet and hand(s), the area of stability, and the vertical projection of the center of mass onto the ground.

3.7

Explain why the center of mass of a standing child, as determined by the fraction of body height over the soles of the feet, is higher ($\sim$5%) than in adults [16].

3.8

(a) WhyStability is a tripod stable?

(b) Explain why a moving six-legged creature (an insect) can always be stable if it lifts and moves three feet at a time, while keeping its other three feet on the ground [117].

(c) Explain how a four-legged animal (a horse) can move, albeit slowly, always maintaining stability, by keeping three of its feet on the ground at all times. Is this type of motion normal for a four-legged animal? Why?

(d) The two-legged motion of humans cannot take advantage of the stability of a tripod. Still, standing barefoot on one foot or both feet can sometimes approximate a tripod. Explain this and why such tripods are fairly unstable.

3.9

Stability ThrowingHow does the force due to the biceps brachi cause rotary motion in the different throwing phases in Fig. 3.47a due to the force component normal to the lower arm [48]? Show in addition that in the early phases when $\theta >90^{\circ }$ there is a component of this force that tends to stabilize the lower arm with the upper arm and that in the later phases when $\theta <90^{\circ }$ there is a component that tends to destabilize it, with the force tending to dislocate the lower arm. Use force diagrams to illustrate each of these points.

3.10

Friction Coefficient of friction KneesThe normal force on the head of the femur in the knee is 300 N. Find the frictional force for a normal ($\mu =0.003$) and arthritic ($\mu =0.03$) joint.

3.11

Gymnastics Slipping Friction Coefficient of frictionWhat would happen if the coefficient of friction between a hand and the vault were too small during vaulting in gymnastics?

3.12

Pulling, pushingWhen we push a heavy object, why do we usually lean into it?

3.13

Sprinting Running Slipping Friction Coefficient of frictionThe accelerating sprinter described in this chapterPulling, pushing pushes off with 560 N force. If the net downward force on the foot on the ground is $3\,m_{\mathrm {b}}g$ for this 70 kg sprinter, what is the minimum coefficient of friction needed to prevent slipping? Under what conditions is this possible?

3.14

Use the data from Chap. 1 to show that the moments of inertia of a normally standing person about his or her center of mass are roughly the following, about these axes [66]:

(a) 1 kg-m$^{2}$ about a vertical axis.

(b) 11 kg-m$^{2}$ about the axis in the transverse plane that is normal to the vertical axis.

(c) 11 kg-m$^{2}$ about the axis in the mid-sagittal plane that is normal to the vertical axis.

(d) If a person curls up into a ball the moments of inertia about all three axes will be roughly the same. Estimate them.

3.15

Calculate the moments of inertia of a standard man about the three normal axes though his center of mass—described in Problem 3.14, using each of the models in Problem 1.32.

3.16

(a) For each limb and limb segment inRadius of gyration Table 1.9, show that the sum of the squares of the radius of gyration about the center of mass and the distance of the proximal end from the center of mass equals the square of the radius of gyration about the proximal end. (All distances can remain normalized by the segment length.)

(b) Repeat this for the distal end.

(c) Why is this so?

3.17

Sprinting RunningUse the parallel axis theorem to calculate the moment of inertia of a sprinter’s leg about theHip hip axis, as in (a) Fig. 3.83. The moments of inertia of the upper leg (thigh), lower leg (calf), and foot about their respective centers of mass are 0.1052, 0.504, and 0.0038 kg-m$^{2}$, these centers of mass are, respectively, 0.30, 0.45, and 0.53 m from the hip rotation axis, and these segments, respectively, have masses 7.21, 3.01, and 1.05 kg.

Fig. 3.83

Sprinting RunningDetermination of the moment of inertia of a sprinter’s leg about theHip hip axis (based on [66].) For Problem 3.17

3.18

Knees Ankles Hip WalkingHow do the leg joint angles defined in Fig. 3.1 differ from those in our discussion of walking and those defined in the caption to Fig. 3.55?

3.19

Multisegment modelingRelate the angles defined in Problem 2.38 for the multisegment model of Fig. 2.58 to the joint angles.

3.20

Refer to Fig. 3.5 and describe how activating eitherMuscles muscle 5 or 6 along with muscles 2 and 4 affectsHip hip andKnees knee flexion and extension. Characterize each in terms of an effective redistribution of torque between the hip and knee.

3.21

Knees HipRefer to Fig. 3.5 and describe the ways to activate legMuscles muscles to extend the knee without applying any torque to the hip. What are the advantages of each?

Walking and Running

3.22

Walking RunningUse Fig. 3.10 to determine the fraction of time that both legs, one leg, and neither leg is on the ground during:

(a) Walking

(b) Running.

3.23

WalkingSketch the direction of normal forces on the ground-bound foot during the stages of stance. (Hint: At midstance the normal force is vertical, and before and afterwards that foot is decelerated and accelerated respectively and the body center of mass changes.)

3.24

(a) UseWalking the pendulum model of walking to determine how fast someone walks. Every half period the person takes a step that corresponds to the arc length the foot (at the end of the leg of length L) that traverses $30^{\circ }$ during the step (so the step length is $(\pi /6)L$). (This is approximately the maximum swing angle for a fast walk.) (What are you assuming about the swing phase in the gait cycle?)

(b) Calculate the pendulum frequency and this walking speed for a 2 m tall adult and a 1 m tall child. Do the ratios of their frequencies and speeds make physical sense? Explain. (You can use the model that assumes constant leg linear density.)

3.25

Walking Allometric rulesShow that it is reasonable that leg length L scales as $m_{\mathrm {b}}^{1/3}$, and therefore the walking speed scales as $m_{\mathrm {b}}^{1/6}$, as in Table 1.13.

3.26

WalkingObtain the moment of inertia about the center of mass of a leg of mass $m_{\mathrm {leg}}$ and length $L_{\mathrm {leg}}$ with constant linear mass density by:

(a) Changing the integration limits in (3.30)

(b) Using the results of (3.30) and the parallel axis theorem.

3.27

Walking Radius of gyrationCompare the moments of inertia of the whole leg as determined by the constant linear density model, (3.30) and the more refined model (result after (3.34)), with that determined from the total leg radius of gyration given in Table 1.9. (Use the parallel axis theorem.)

3.28

WalkingHow can using a walking hiking stick help you when hiking in regard to:

(a) going uphill,

(b) going downhill,

(c) improving yourStability stability (center of mass), and

(d) preventingSlipping slipping?

3.29

Walking RunningWhy can it be easier to walk and run on wet sand than on dry sand?

3.30

WalkingEstimate what fraction of the change in body angular momentum due to leg motion is canceled by swaying your arms out of phase with your legs when you walk. Assume standard parameters for your arms and legs, that they are straight and each has a uniform mass density per unit length:

(a) First assume that $\mathrm{d}\theta /\mathrm{d}t$ is equal in magnitude and opposite in sign for the legs and arms.

(b) Now assume that the forward displacements traversed by the foot and hand $L\mathrm{d}\theta (t)/\mathrm{d}t$ are equal in magnitude and opposite in sign, where L is alternately the leg or arm length.

3.31

RunningEstimate the vertical angular momentum gained by your body with each stride (ignoring the motion of your arms and torso) for a 3.4 m/s running speed.

3.32

Climbing; ascending, descending stairs StairsEach step in climbing a staircase can be modeled in two parts. First, one foot is placed on the next step and, second, that foot is used to propel the body up. Assume that the center of mass is raised an insignificant amount in the first part, so the whole change in the center of mass occurs in the second part of the step. Also, assume that the durations of both parts of the step are the same. Consider a staircase with 13 steps, each 20 cm high. It takes 3.6 s to go up the staircase “quickly” and 6.0 s to go up “slowly.” Calculate the average total (not net) vertical force in the second part of the step for each case, in terms of the body weight $W_{\mathrm {b}}$. (You can ignore horizontal motion.)

3.33

Knees Ankles Hip RunningDetermine the total range of motion of thigh, knee, and ankle for each example of running in Table 3.6.

3.34

Achilles tendon(a) A 70 kg personRunning runs at a speed of 4.50 m/s. If 100 J of kinetic energy is lost each time a foot touches the ground (Fig. 3.28), what is the speed after stage (a) (if it was 4.50 m/s just before stage (a)). (Ignore vertical motion and potential energy changes throughout this problem.)

(b) How much energy needs to be supplied by the body in the acceleration phase (stage (c)) to account for the loss when the foot earlier hit the ground, if the 93% of the 35 J which is stored in the Achilles tendon and the 80% of the 17 J which is stored in flattening the foot arch in stage (a) are both returned in stage (c) (and nothing else is)?

(c) IfRunning the runner takes 3 steps per second and the runner’s muscles are 20% efficient in converting energy into the mechanical work of running, how much extra energy is used by the body per hour when it is running? Express your answer in kcal/h, where 1 kcal $=$ $1\times 10^{3}$ cal and 1 cal $=$ 4.184 J.

3.35

Sprinting RunningExplain why sprinters start a racePulling, pushing pushing off from blocks.

3.36

Sprinting RunningAfter how many strides does the accelerating sprinter reach 90% of her final speed? What distance does this correspond to (in m)?

3.37

Sprinting RunningThe final speed attained of the accelerating sprinter refers to the upper body and one leg. If the other leg can be considered to be still, what is the speed of the sprinter’s center of mass after acceleration?

3.38

Sprinting RunningWe assumed that thePulling, pushing pushoff leg of the accelerating sprinter does not contribute any kinetic energy in (3.38)–(3.47). Let us say that the center of mass of that leg is moving at half of the speed of the upper body and other leg:

(a) Is this reasonable? Why?

(b) How would this change (3.38)–(3.47)?

(c) How would it change the numerical value of the calculated final speed?

(d) How would it change the number of strides needed to reach 90% of the final speed?

3.39

Sprinting RunningWe assumed that thePulling, pushing pushoff leg of the accelerating sprinter does not contribute any kinetic energy in (3.38)–(3.47). Let us say that this leg rotates while this foot is on the ground, so the top of it moves at the same speed as the rest of the body, but the bottom of it is still. Assume this leg is straight and of uniform linear density. What fraction of the kinetic energy of the leg is lost as it changes from a translating to a rotating leg?

3.40

Sprinting RunningDo parts (b)–(d) of Problem 3.38 for the case posed by Problem 3.39.

3.41

Sprinting RunningPlot the data points in Table 6.34 as average speed in miles/hr versus distance in m (with this axis plotted on a log scale), and fill in the curve between these points either by hand or by curve fitting.

3.42

Running(a) Use the data in Table 6.34 to model the average world record running speeds for men, $v_{\mathrm {av}}$ (in mi/hr) for distances D (in m) longer than 400 m as $v_{\mathrm {av}}=K/D^{n}$ [16].

(b) Estimate what the world record time for running 4,000 m would be expected to be.

(c) Estimate how fast you would expect the speed to be in the middle of that race? (If you are assuming that the speed is constant during the race, why would you think this is a reasonable assumption?)

(d) Compare the 2014 men’s world record with those in 1957 ($K = 29.66$, $n= 0.089$) and 1982 ($K = 29.91$, $n = 0.085$). (Plot all on the same set of axes and discuss.)

(e) Why are shorter distances not used in this type of analysis?

3.43

Running Allometric rulesShow that the maximum (horizontal) running speed of mammals is expected to be roughly independent of mass by using the following approach [95, 130]. Assume that the maximumMuscles muscle force is proportional to its cross-sectional area, which is $\propto L^{2}$ for characteristic dimension L. The body mass is proportional to the volume, and so $m_{\mathrm {b}}\propto L^{3}$. Proceed as follows:

(a) The work done by leg muscles in each stride is the muscle force $\times$ the muscle contraction distance. Show that this scales at $L^{3}$.

(b) The kinetic energy of that limb is $I(\mathrm{d}\theta /\mathrm{d}t)^{2}/2$, where I is the moment of inertia of the leg and $\mathrm{d}\theta /\mathrm{d}t$ is its angular speed, which is the running speed v divided by the leg length. Show that this energy scales as $L^{3}v^{2}$.

(c) By equating these, show that v is independent of L and therefore $m_{\mathrm {b}}$.

3.44

Running Allometric rulesShow that the maximum uphill running speed of mammals should decrease with linear dimension L, as 1 / L, and therefore with mass $m_{\mathrm {b}}$ as $1/m_{\mathrm {b}}^{1/3}$ [95, 130]. Do this by equating the power available from theMuscles muscles to the power required to work against gravity. (Hint: The power available from muscles is the work done by leg muscles per stride ($\propto L^{3}$, from Problem 3.43) divided by the time per stride (which is the leg length divided by the speed, and Problem 3.43 shows the speed is independent of L). The power needed to work against gravity is the body weight times this uphill running speed.)

Jumping

3.45

Vertical jump JumpingNeglect air resistance and muscle atrophy and assume the same upward normal reaction force as on Earth:

(a) If someone can increase her center of mass by 0.7 m in a vertical jump on Earth, how high could she jump on the moon?

(b) If an athlete can long jump 25 ft on Earth, how far could she long jump on the moon? (Assume her takeoff angle is $30^{\circ }$.)

3.46

(a) A personKnees Ankles Hip Vertical jump Jumping tries to touch the highest point possible by fully extending his/her arm during a vertical jump. You are told that before takeoff, the ankle and knee joints are at peak flexion, with angles 85$^{\circ }$ and 90$^{\circ }$ and that after takeoff and at peak height, these angles are 149$^{\circ }$ and 178$^{\circ }$. From the context of the information given, how are these angles being defined? Are these definitions the same as those in Fig. 3.1?

(b) Make a sketch of the body at these positions, which should include the ankle, knee, and hip pivot points (in this 2D sketch) and these angles.

(c) Where do these angles fall within the ranges of motion of the two joints?

3.47

(a)Muscles Vertical jump Jumping During a vertical jump the center of mass of a 70 kg person is 0.65 m from the ground in the crouch phase and 1.05 m at takeoff. During the extension phase the average (total) force exerted by the floor (summed on both feet) is 1,600 N. (Remember that part of this counters the body weight and the remainder—the net vertical force—counters the forces due to muscles.) When necessary, assume that this force is constant during the extension phase. Find how high the center of mass rises during free vertical flight, the speed at takeoff, and the (temporal) duration of the extension phase.

(b) About 43% of the mass of the average person is muscle and essentially all of it is skeletal muscle mass. This average person in (a) doesWeightlifting, training weight training, which increases his body mass by 5 kg. Assume all of it goes into muscle proportionately throughout the body and that the total vertical force is proportional to the total body muscle mass. How high does his center of mass rise now during a vertical jump?

(c) Repeat (b) (starting with the person in (a)) if the “weight training” instead leads to adding only 10 kg of muscle.

(d) Repeat (b) (starting with the person in (a)) if the “weight training” instead leads to adding only body fat—with no change in muscle mass.

(e) Find how high the center of mass rises in (a) if the person takes a wonder drug that decreases the extension phase by 0.05 s.

(f) On the moon, how high does the person in (a) rise in a vertical jump (from a hard floor)? Assume $g_{\mathrm {Moon}}=g_{\mathrm {Earth}}/6$ and that the muscles function exactly as on Earth.

3.48

Vertical jump JumpingRedo Problem 3.47b if the person now does not gain weight, but instead either increases hisMuscles muscle mass fraction from 43 to 53% or decreases it to 33%.

3.49

(a)Muscles Vertical jump Jumping If the person does a vertical jump jump from a rubber pad or sand instead of a hard floor the center of mass would rise less. Why?

3.50

Pole vaultThe world record pole vault (6.16 m) exceeds that calculated here (5.4 m). Compare this world record to the calculation assuming world class speed, 10.4 m/s, and $h_{\mathrm {min}}=0.02$ m. Why are they different?

3.51

High jump Jumping RunningIn a high jump the athlete takes a running start and then hurls himself over a horizontal pole. (In the older, straddle high jump (face down when over the bar), the athlete’s center of mass is $\sim$ $150$ mm over the bar, while in the newer Fosbury flop method (face up when over the bar), the center of mass is slightly under the bar. Assume here that the center of mass needs to clear the bar by 100 mm during a successful high jump ($H_{3}$ in Fig. 3.41)):

(a) If a 70 kg athlete’s initial center of mass is 0.9 m high ($H_{1}$), how high can the bar be for a successful jump ($H_{1}+H_{2}-H_{3}$) if the athlete runs at 7.0 m/s and all of his initial kinetic energy (corresponding to motion in the horizontal direction) is converted into kinetic energy corresponding to motion in the vertical direction and then into potential energy?

(b) Since the world record high jump is 2.45 m (in 2014), how much of the initial kinetic energy could not have gone into kinetic energy associated with vertical motion?

(c) Measurements show that when 76 kg athletes run 6.7 m/s and high jump over a bar that is 2.0 m high, their horizontal speed over the bar is 4.2 m/s. How much energy is still not accounted for? What happened to it? (Note that the total kinetic energy (in two-dimensions, x and z) is the sum of that for horizontal and vertical motion $=mv_{x}^{2}/2+mv_{z}^{2}/2$.)

3.52

High jump JumpingIn a high jump using the Fosbury flop method (face up when over the bar, as in Fig. 3.41), the center of mass can be slightly below the bar. Find $H_{3}$ if the athlete height is 1.96 m, $H_{1}=1.40$ m, $H_{2}=0.97$ m, and the height of the bar is 2.30 m (as for the jumper Dwight Stones, [66]).

3.53

High jump JumpingFor the high-jumper modeled with a two-segment model of the Fosbury flop in Fig. 3.84, calculate how much lower the center of mass of the jumper is than the bar for positions (a)–(e). How much lower is it for position (c)?

Fig. 3.84

Simple one-segment model of a high jump using the Fosbury flop method for a person with height H, with the body modeled as two segments of equal length, normal to each other. In (a)–(e), the top of the modeled body is always $0.1\,{H}$ above the bar (which is the dot). This is a reasonable assumption for (c), because the center of the person’s chest is high enough for her to clear the bar. For Problems 3.53 and 3.54

3.54

High jump JumpingAssume the two segments of the Fosbury flop jumper in Fig. 3.84 are at an obtuse angle, rather than the right angle shown in the figure. (This is more realistic.) Also assume that the general orientation of the jumper is otherwise the same in each part of the figure. Find the maximum angle for which the center of mass of the jumper will always remain below the bar.

3.55

(a)Long jump Jumping The length of the long jump has three parts, the takeoff distance ($L_{1}$, the center of mass precedes the foot at takeoff, take as 0.24 m here), the flight distance ($L_{2}$, the distance the center of mass travels during flight), and the landing distance ($L_{3}$, the distance the heel lands in the sand in front of the center of mass, take as 0.53 m), as in Fig. 3.42. Calculate the lengths of world-class male and female long jumps with speeds at takeoff of 9.8 and 8.6 m/s, respectively, and a takeoff angle of $20.0^{\circ }$.

(b) We have ignored drag here, as well as the difference in the height of the center of mass at takeoff and landing. How are the results in (a) affected if the center of mass at takeoff is 60 cm above that during landing?

3.56

Drag Long jump JumpingA 70 kg world-class long jumper accelerates to a speed of 10.5 m/s and then jumps the longest distance possible:

(a) If drag due to air resistance is neglected, show that this occurs at a $45^{\circ }$ takeoff angle, and find the length of this longest possible long jump. (Assume that the long jump distance is the same as the horizontal distance traveled by the center of mass. (They can be slightly different because of the different takeoff and landing arrangements of the body, as is addressed in Problem 3.55)).

(b) Compare this distance to the world record of 8.95 m (in 2014), and qualitatively account for any differences between this record jump and your calculated value. (Consider that many long jumpers take off at an angle closer to $20^{\circ }$.)

3.57

Drag Long jump JumpingIf drag due to air resistance is neglected, a projectile, initially at ground level, travels farthest with a $45^{\circ }$ takeoff angle. Including air resistance the takeoff angle is closer to $\sim$ $35^{\circ }$. Given this, does it makes sense that many long jumpers take off at an angle closer to $20^{\circ }$?

3.58

Jumping Muscles Allometric rulesShow that the maximum jumping height of the center of mass of mammals is expected to be roughly independent of mass. Do this by first showing the acceleration a due to muscles varies as 1 / L, for characteristic mammal linear dimension L. Then show that the launch speed, $v=\sqrt{2as}$—where s is the vertical distance traveled during acceleration—is independent of L and so the height of the jump $v^{2}/2g$ is expected to be independent of mass. Use the same scaling rules for the maximum muscle force and body mass as in Problem 3.43. (How does s scale with L?)

Other Motions

3.59

GymnasticsExplain why circus performers doing tightrope and high-wire actsCarrying carry a long horizontal pole [16].

3.60

Skating(a) A figure skater, rotating at an angular frequency of f (in revolutions per s, which can be $\sim$ $0.6$ revolutions/s) with outstretched arms, rotates even faster after pulling in her arms. Why?

(b) The moment of inertia of this person changes by a factor $\alpha$ by this pulling action. What is the skater’s new rotation speed?

(c) Estimate this factor. (You can use Tables 1.6–1.9 and Problems 1.32 and 1.33 for guidance.)

(d) Does this maneuver increase the kinetic energy of the skater? Is so, where does this extra energy come from?

3.61

Explain why your pumping motions during your forward and backward motions on a swing enable you to go higher and higher. (Show how both types of pumping decrease your moment of inertia about the rotation axis and that there are no external torques.)

3.62

PushupsDescribe the joint action of each joint in the dip phase of a pushup.

3.63

(a)Pulling, pushing Pushups Sketch four ways of doing a pushup with your hands pushing up on a horizontal surface (to improve your arm strength): (i) with yourKnees knees (and not feet) and hands on level ground, (ii) with your feet (and not knees) on the ground and your hands on the fourthStairs stair on a nearby upstairs stairway, (iii) with your feet on the ground and your hands on the second stair on the stairway, and (iv) with your hands and feet both on level ground (which is the conventional pushup). Show that in each case the normal force that is exerted on your hands in equilibrium has magnitude $N = m_{\mathrm {b}}g (L/H)$, for body weight $m_{\mathrm {b}}g$ and horizontal distances from your hands and the center of mass to the rear point of contact (knees or feet) of H and L respectively.

(b) Show that you can successfully do a pushup only when you exert a force $N > m_{\mathrm {b}}g (L/H)$.

(c) Show that a larger N is successively needed from cases (i) to (iv), so the pushup becomes successively harder to do and your workout becomes more meaningful. (Make reasonable assumptions about these distances as needed.)

3.64

BoatingExplain why the two people in a canoe paddle on opposite sides of the canoe when they want to go straight and why they paddle on the same side when they want to turn (and do so on the side opposite from the turning direction).

3.65

Diving Somersaulting BouncingA platform diver bounces on the platform and after takeoff does a triple somersault. Describe the motion of the center of mass during this complicated maneuever.

3.66

Skiing Drag Friction Coefficient of friction(advanced problem) How far does a downhill skier travel as a function of time [135]?

(a) Show that you can rewrite (3.138) as

$$\begin{aligned} m_{\mathrm {b}} v \frac{\mathrm{d}v}{\mathrm{d}s} + \frac{1}{2}C_{\mathrm {D}}A \rho v^{2}= m_{\mathrm {b}}g \sin \alpha - \mu _{\mathrm {k}} m_{\mathrm {b}}g \cos \alpha \end{aligned}$$

(3.158)

and then as

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}s}\left( \frac{1}{2} v^{2}\right) + \frac{\rho A C_{\mathrm {D}}}{m_{\mathrm {b}}}\left( \frac{1}{2} v^{2}\right) = g (\sin \alpha - \mu _{\mathrm {k}} \cos \alpha ). \end{aligned}$$

(3.159)

(b)Drag Show that multiplying both sides by the integrating factor $\exp ( \rho A C_{\mathrm {D}}s/m_{\mathrm {b}})$, gives

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}s}\left( \frac{1}{2} v^{2} \exp ( \rho A C_{\mathrm {D}}s/m_{\mathrm {b}})\right) = g (\sin \alpha - \mu _{\mathrm {k}} \cos \alpha ) \exp ( \rho A C_{\mathrm {D}}s/m_{\mathrm {b}}). \end{aligned}$$

(3.160)

(c) NowSkiing show that integrating this equation gives

$$\begin{aligned} v(s)^{2} = \frac{2 m_{\mathrm {b}} g (\sin \alpha - \mu _{\mathrm {k}} \cos \alpha ) }{ \rho A C_{\mathrm {D}}}(1- \exp ( -\rho A C_{\mathrm {D}}s/m_{\mathrm {b}})) , \end{aligned}$$

(3.161)

Skiingassuming that v = 0 when s = 0, meaning the skier starts at rest at s = 0, or, using (3.139),

$$\begin{aligned} v(s) = v_{\mathrm {terminal}} \sqrt{1- \exp ( -\rho A C_{\mathrm {D}}s/m_{\mathrm {b}})} \end{aligned}$$

(3.162)

or

$$\begin{aligned} \frac{\mathrm{d}s}{\mathrm{d}t} = v_{\mathrm {terminal}} \sqrt{1- \exp ( -\rho A C_{\mathrm {D}}s/m_{\mathrm {b}})}. \end{aligned}$$

(3.163)

(d)Friction Coefficient of friction Drag After putting all the s terms on one side of the equation and all t terms on the other and integrating, show that the distance traveled s (after a time t) and t are related by

$$\begin{aligned} t=\frac{1}{k v_{\mathrm {terminal}}} \ln {\left( \frac{1+z}{1-z}\right) }, \end{aligned}$$

(3.164)

where $k = \rho A C_{\mathrm {D}}/ m_{\mathrm {b}}$ and $z^{2} = 1 - \exp (-ks)$.

3.67

Skiing Drag FrictionUse the results of Problem 3.66 to show that for the example in the text where the downhill skier attains a terminal velocity of 25.1 m/s, the skier travels 2,500 m in 108 s [135].

3.68

Skiing Drag FrictionShow that (3.161) in Problem 3.66 reduces to the expected results when there is no friction or drag.

3.69

Skiing(advanced problem) This problem determines the path of a downhill skier that takes the minimum time [88]. With the lateral coordinate x and the downhill coordinate on the plane y, which increases down the slope, and the downhill angle to the horizontal $\alpha$, the skier decides to start at rest at $x=0$ and $y=0$ and end at $x=\beta y_{\mathrm {f}}$ and $y=y_{\mathrm {f}}$. The distance along the path is called s, so $\mathrm{d}s= \sqrt{1+y'^{2}}\,\mathrm{d}x$ where $y'=\mathrm{d}y/\mathrm{d}x$.

(a) Show that the skier has speed $v=\sqrt{2g'y}$, where $g'=g \sin \alpha$. (This depends on y and not x. We will be assuming that the skier can change directions at any y without loss of speed, and will ignore drag and friction.)

(b) Show that skier arrives at the destination at time $t=\int _{0}^{y_{\mathrm {f}}} \sqrt{(1+ y'^{2})/2g'y}\ \mathrm{d}x$.

(c) Variational analysis shows that t is a minimum if $F- y' \partial F/ \partial y'$ is a constant where $F=\sqrt{(1+ y'^{2})/2g'y}$. Show that this means that $y(1+ y'^{2})=C$, a constant.

(d) By substituting $y'=\cot \theta$, show that $y=(C/2)(1-\cos 2\theta )$.

(e) Show that $\mathrm {d}x/\mathrm {d}\theta =(1/y') \mathrm {d}y/\mathrm {d}\theta = C(1-\cos 2\theta )$.

(f) With $C=2A$, $2\theta =\phi$, and $R=A/2 g'$, show that $x=R(\phi -\sin \phi )$ and $y=R(1-\cos \phi )$. This is cycloid obtained by a point on a circle of radius R rolling on a straight line.

3.70

Skiing(a) You are going downhill at speed v in a cart on a track that enters a turn with radius of curvature R or on skis and enter a trough with radius of curvature R, and move so the center of mass (of you and the cart or of you, respectively) moves h closer to the center of radius ($h\ll R$). If your angular momentum does not change during this turn, show that you have a fractional increase in speed of h / R and kinetic energy of 2h / R [88, 104].

(b) Explain why there are no external torques during this motion, so the angular momentum does not change.

(c) Where does the increase in kinetic energy come from?

(d) Sketch where the center of curvature is relative to the slope is both cases, before and during the turn.

3.71

Skiing(a) When a downhill skier raises her center of mass by 0.5 m at the bottom of a trough with radius of curvature 4 m, find the fractional increase of her kinetic energy.

(b) If she has a speed of 10 m/s, find the normal force she feels during this maneuver, expressed in terms of her body weight.

3.72

DragA Nordic ski jumper takes off with a horizontal speed of 15 m/s from a tower that is 10 m above the slope that descends at a $45^{\circ }$ angleSkiing Drag Lift.

(a) How far down the slope does the skier land, measured along the slope from the base of the tower? Ignore the effect of air.

(b) If the effect of air is included, a 70 kg jumper with a frontal area of 0.7 m$^{2}$ will land 65.9 m down the slope (using an air density of 1.007 kg/m$^{3}$, $g = 9.81\,$m/s$^{2}$, and drag and lift proportional to the square of the speed and $C_{\mathrm {D}}= 1.78$ and $C_{\mathrm {L}}= 0.44$) [37]. How does your answer compare to this and what does this mean?

3.73

A skier jumps from a ramp with speed of 15 m/s$^{2}$ at a $45^{\circ }$ angleSkiing Drag Lift. If the skier lands at the same height as she takes off (so the snow line is horizontal and the ramp height is negligible), how far is the jump and how long does it take? Analyze this first by ignoring drag and lift then by including only drag (using the information in Problem 3.72 and Table 3.12). How important is drag?

Throwing Balls and Other Objects, and Ball Flight

3.74

Use (7.11) to show that $Re \simeq 640\,vd$ in air.

3.75

CricketShow that $Re \sim 140{,}000$ for a cricket ball (7.2 cm diameter) traveling at 30 m/s.

3.76

Find the Reynolds number for the following balls traveling a 30 m/s:

(a)Soccer soccer (diameter of 8.7 inches, 22.1 cm),

(b)Tennis tennis (2.5 inches, 6.35 cm),

(c)Golf golf (1.68 inches, 4.27 cm),

(d) squash (1.6 inches, 4.07 cm),

(e) ping pong (table tennis)Ping pong, table tennis (1.5 inches, 3.82 cm) [36].

(f) How do these compare with the critical value?Balls Drag

3.77

Explain how energy generated by the rotation about the shoulder can be transferred to rotation of the elbow when throwing a ball, by treating the upper arm and the forearm as a double pendulum.

3.78

Football Baseball Throwing Pitching Balls(a) Use the one-segment model in this chapter that predicts you can throw a baseball at 45 mph to determine how fast can you throw a football. A National Football League (NFL) football must weigh between 14 and 15 oz (and have a mass between 0.40 and 0.43 kg). Is this consistent with an NFL quarterback throwing a ball at 68 mph? Does this make sense?

(b) Would it make sense if you modified the one-segment ball throwing model so the baseball is released at 100 mph and then used it for throwing a football?

3.79

Baseball Throwing Pitching BallsIn the one-segment model for throwing a ball in this chapter, what should the angle of the upper arm be relative to the vertical so that the velocity vector of the released ball is horizontal (with a release at $\theta =\pi /4$)?

3.80

(a)Baseball Throwing Pitching Balls Modify the one-segment model of throwing a baseball so that the whole arm is always straight and is rotated about the shoulder by theMuscles deltoid muscles.

(b) How fast is the ball released, assuming 3 in diameter deltoid muscles and the other assumptions in the final model in this chapter?

3.81

(advanced problem) Show that when $\alpha _{\mathrm {1}}$ and $\omega _{\mathrm {1}}$ are zero the two-segment model torque equation for segment 2 (3.100) reduces to that for a one-segment model (3.68) for segment 2. Why should this be so?

3.82

(advanced problem) Show that when $m_{\mathrm {2}}$ and $I_{\mathrm {2p}}$ are zero the two-segment model torque equation for segment 1 (3.101) reduces to that for a one-segment model (3.68) for segment 1. Why should this be so?

3.83

(advanced problem) Show that when $\theta _{\mathrm {21}}$, $\omega _{\mathrm {21}}$, and $\alpha _{\mathrm {21}}$ are zero the two-segment model torque equation for segment 1 (3.101) reduces to that for a one-segment model (3.68) for the rigid, straight assembly of segment 1 and segment 2. Why should this be so? (You will need to use the parallel axis theorem (3.27), which relates the moment of inertia about the axis through the center of mass to that about another axis.)

3.84

Baseball Throwing Pitching BallsHow do the torques we calculated in our pitching model compare to those in Figs. 3.50 and 3.51? Should they be comparable?

3.85

Baseball Throwing Pitching BallsEstimate the relative final angular rotation speeds of the elbow and wrist in throwing a ball by using Fig. 3.49, and then estimate the relative contributions to the speed of the pitched ball from the motion of the lower arm and the wrist. Assume the ball is a distance of half of the length of the hand from the wrist joint.

3.86

Baseball Throwing Pitching Balls MusclesEstimate the mass of the internal rotator muscles (in kg) in the shoulders of very large people, who are good baseball pitchers with rapid humerus rotation during pitching. These muscles (and their volumes in cm$^{3}$) are: pectoralis major (676), latissimus dorsi (550), anterior deltoid (one third of the 793 total), teres major (231), and subscapularis (319), and the muscle density is 1.06 g/cm$^{3}$ [121]Pitching Humerus Shoulders Muscles.

3.87

Can theBaseball Throwing Pitching Balls Muscles muscles in the shoulder account for all of the power generated by the shoulder joint during a baseball pitch? Assume that the work done during shoulder rotation during the pitch is 132 J during the acceleration phase that lasts 0.034 s.

(a) Find the average shoulder joint “kinetic energy” power (or kinetic power).

(b) Assume that the maximum power generated by muscles is 250–500 W/kg and the muscle mass is as determined in Problem 3.86. (This mass is also given in the solutions.) How many times larger is the average shoulder kinetic power than the maximum muscle power?

(c) The muscle power given in (b) is for isotonic muscle conditions (see Chap. 5). Prestretching may increase this by a factor of 1.75. How many times larger is the average shoulder kinetic power than this new maximum muscle power?

(d) What can you conclude about the shoulder muscles being responsible for humerus rotation during baseball pitching [121]?Pitching Humerus Shoulders Muscles

3.88

Football Throwing Pitching BallsTo throw a football well, you need to grasp it tightly. A regulation National Football League football has an internal gauge pressure of 12.5–13.5 psi, weight of 14–15 oz, circumference at the center of $\simeq$22 inches, and an $\simeq$11 inch tip to tip distance. Use Problem 7.14 to estimate the range of pressures and forces a quarterback’s hands need to exert to grasp a football well, assuming his hand makes contact over 2.5 inch$^{2}$ of the ball.

3.89

Football Throwing Pitching Balls(a) A football is found to be inflated to a gauge pressure 2 psi below the allowed minimum. The football may have been inflated correctly by officials, and then deflated (by releasing internal air) by someone trying to help a quarterback with weak or small hands. Estimate the pressure and force that needs to be exerted by the quarterback’s hands to grasp this football, assuming his hand makes contact over 2.0 inch$^{2}$ of the ball. (Refer to Problem 3.88.)

(b) A reporter says that the pressure in such deflated footballs could be determined by weighing them, to see if they are still within the official limits. Explain why this will not work. Devise a better way to determine the pressure.

(c) With constant external pressure of 14.7 psi the internal gauge pressure of the footballs could decrease because the ambient became colder. How cold would it need to be for the gauge pressure to have decreased by 2 psi, if they were inflated correctly when the ambient temperature had been 70 $^{\circ }$F?

3.90

Football Throwing Pitching BallsWould using the deflated footballs of Problem 3.89 help running backs grasp onto the ball better, so they could avoid fumbling (losing) the ball?

3.91

Baseball Throwing Pitching BallsA baseball pitcher throws a ball with initial velocity vector component $v_{x,0}$ in the forward direction and $v_{z,0}$ in the vertical direction. (Ignore ball deceleration due to drag):

(a) Show that after a time t, its coordinates are: $x=v_{x,0}t$ and $z=v_{z,0}t-gt^{2}/2$.

(b) After it travels a distance d in the x direction, show that the ball seems to have fallen a distance $gd^{2}/2v_{x,0}^{2}$ from its initial trajectory.

(c) Show that the ball always stays in the $y=0$ plane.

3.92

BallsIf the initial velocity vector component of a ball is $v_{x,0}$ in the forward direction and $v_{z,0}$ in the vertical direction, show that it travels a distance $2v_{x,0}v_{z,0}/g$ (which is called its range), which equals $v_{0}^{2}\sin (2\alpha )/g$, for initial speed $v_{0}$ and angle $\alpha$.

3.93

Balls Striking, hitting, punching Kicking Throwing(a) Show that a ball travels farthest when released at a $45^{\circ }$ takeoff angle, and when it is released at speed v, it travels a distance of $v^{2}/g$. (DragThis is an overestimate because air drag is omitted. When drag is included, the best angle is really $35^{\circ }$. Also, we are ignoring any difference in the heights of the release point and the ground. This will be examined later in the problems.)

(b)Baseball How far does the baseball travel when thrown at 90 mph?

(c) If the ball is wet and weighs 7 oz instead of 5 oz, how far will it travel? Assume the ball leaves the hand with the same linear momentum as the dry ball does in part (b).

3.94

Balls Striking, hitting, punching Kicking Throwing(a) A ballKicking kicked or thrown travels the farthest with takeoff angle 45$^{\circ }$. For a given angle $\alpha _{1} < 45^{\circ }$, find the angle $\alpha _{2} > 45^{\circ }$ at which the ball travels the same (lesser) distance. Ignore drag.

(b) What are the travel times at each angle? Are they equal?

(c)Drag With drag included, do you expect the travel distances at these two angles still to be equal? Why?

3.95

Football(a) When a punterKicking kicks (punts) a football to the opposing team, the goal is usually to kick it the farthest, with one constraint: His “covering” teammate, who is next to the ball and startsRunning running in the direction of the ball when it is punted, should not arrive after the opposing player (punt returner) canCatching catch the ball, so he can tackle the returner before he can run with (return) the ball. The kicker kicks the ball an initial speed v at takeoff angle $\alpha$ and his teammate runs at speed u (essentially staying under the traveling ball). For a given v and u ($<v$), find the optimal takeoff angle, and the punt distance and travel (hang) time at this angle. (DragDrag significantly changes these results [16], but ignore it here.)

(b) Find the optimal angle, punt distance (in yards) and hang time (in s) for a kickoff speed of 55 mph, if the covering player can run 40 yards in 4.3 s.

3.96

Balls Striking, hitting, punching Kicking ThrowingThe farthest distance an object can be thrown is with a takeoff angle of 45$^{\circ }$, ignoring drag, only if it is thrown from ground height. Otherwise this is just an approximation, as we will now see:

(a) If an object is thrown with an initial speed v and angle $\alpha$, from a height $z=h$ show that at a later time: $x(t)=vt\cos \alpha$ and $z(t)=h+vt\sin \alpha -gt^{2}/2$.

(b) Show that it reaches the ground, $z(t)=0$, after it has traveled a distance $x_{\mathrm {final}}=(v^{2}\cos \alpha /g)(\sin \alpha +\sqrt{q^{2}+\sin ^{2}\alpha })$, where $q=\sqrt{2gh/v}$. (Note that $q^{2}=mgh/(mv^{2}/2)$, which is the initial potential energy divided by the initial kinetic energy.)

(c) By setting $\mathrm{d}x/\mathrm{d}\alpha =0$, show that the ball will travel farthest for an initial angle given by $\tan \alpha _{\mathrm {max}}=1/\sqrt{1+q^{2}}$ and it travels a distance $x_{\mathrm {final,max}}=(v/g)\sqrt{v^{2}+2gh}$.

3.97

Baseball Balls ThrowingA baseball is thrown from a height of 6.5 ft from the ground with an initial speed of 90 mph. Ignoring drag, find how far it travels if:

(a) it is released at $45^{\circ }$, and you ignore the difference in the release height and the ground.

(b) it is released at $45^{\circ }$, and you take into account the difference in the release height and the ground (see Problem 3.96).

(c) it is released at the optimal angle for distance (which you also must find), and you take into account the difference in the release height and the ground (see Problem 3.96).

3.98

Baseball Throwing Pitching BallsA baseball pitcher throws a ball at 90 mph releasing it 6 ft in front of the pitcher’s rubber (the stripe on the pitcher’s mound), at a height of 5 ft. The ball arrives at the plate, which is 60 ft 6 in from the pitcher’s rubber at a height of 3 ft. (Ignore ball deceleration due to drag):

(a) How long does is take the ball to arrive?

(b) What angle does the initial velocity vector of the ball make with the horizontal?

(c) When the ball arrives at the plate, how much does it appear to have fallen relative to the path it would have taken with its initial velocity vector? (Ignore any effects due to ball rotation.)

3.99

Baseball Throwing Pitching BallsWhy are pitchers often tall? Consider where they release the ball and batter reaction times, how fast they can throw (if arm rotation speed were independent of height), drag during flight, and, for those who throw sidearm, the angle the ball can cross the plate.

3.100

Balls Striking, hitting, punching Swinging BaseballA batter hits a baseball so it initially has a speed of 90 mph and travels 45$^{\circ }$ to the horizontal. Find how far it travels and its travel time assuming it travels in vacuum and has the same initial and final heights. Assume a baseball weighs 5 1/8 ounces and has a circumference of 9 1/8 inches (regulation values).

3.101

Balls Striking, hitting, punching Swinging BaseballRepeat Problem 3.100, now for travel in air (at 70 $^{\circ }$F, no humidity, at sea level), by numerically integrating (3.136) and (3.137). Assume $C_{\mathrm {D}}$ is 0.45. Compare your results with those of Problem 3.100.

3.102

Balls Striking, hitting, punching Swinging Baseball DragRepeat Problem 3.101, by using the ballistic table, Table 3.12. (Calculate $b_{1}$ and then determine $b_{2}$ by interpolation in this table to obtain the range (in feet).) Compare your results with those in Problem 3.100. (If you are also doing Problem 3.101 compare your results with those of both problems.)

3.103

Balls Striking, hitting, punching Swinging Baseball DragA batter hits a baseball so it initially has a speed of 90 mph and travels 45$^{\circ }$ to the horizontal. Let us explore how far it travels under several different air conditions, assuming its initial and final heighs are the same, by using the ballistic table, Table 3.12. Assume a baseball weighs 5 1/8 ounces and has a circumference of 9 1/8 inches (regulation values), and $C_{\mathrm {D}}$ is 0.45. Air drag is proportional to the mass density of the air $\rho$, which itself depends on the number density of each constituent (from its partial pressure) times its molecular mass.

(a) First find the range in vacuum. (You cannot use the ballistic table for this. You might want to consult other chapter problems for guidance.)

(b) Now, calculate $b_{1}$ and then determine $b_{2}$ by interpolation in this table to obtain the range (in feet) for the following conditions: (i) 70 $^{\circ }$F, no humidity, at sea level (where atmospheric pressure is 760 Torr = 760 mmHg); (ii) 30 $^{\circ }$F, no humidity, at sea level; (iii) 100 $^{\circ }$F, no humidity, at sea level; (iv) 100 $^{\circ }$F, maximum humidity (water vapor pressure is 49 mmHg, the rest is air), at sea level; (v) 70 $^{\circ }$F, no humidity, in Denver (where atmospheric pressure is 626 mmHg).

(c)Drag How important are drag and these variations in air conditions? As part of this compare your results to those expected for the ball traveling in vacuum.

(d) Why do people say that balls travel well on hot humid days and at high elevations, and poorly early in the baseball season (i.e. the early spring)?

3.104

Soccer Drag BallsDetermine the range and time of flight of a soccer ballKicking kicked with initial speed 28 m/s at angle 50$^{\circ }$ by using the ballistic table, Table 3.12 [37]. The mass of the soccer ball is 0.42 kg and its diameter is 0.22 m. Assume $C_{\mathrm {D}} = 0.2$ (because of the very high Re).

3.105

Soccer Drag BallsA soccer ballKicking kicked at $45^{\circ }$ initially travels at a speed of 60 m/sSoccer Drag. Assume it does not spin.

(a) Show that its Reynolds number exceeds the critical value, at least at the beginning of its flight.

(b) How far would it travel in vacuum?

(c) How far would it travel if throughout the flight the drag coefficient $C_{\mathrm {D}}$ were assumed to be the value 0.45, for smooth balls with a Reynolds number below the critical value?

(c) How far would it travel if throughout the flight the drag coefficient $C_{\mathrm {D}}$ were assumed to be 0.15 (because the Reynolds number exceeds the critical value)?

3.106

Golf Drag Balls Striking, hitting, punchingDetermine the range and time of flight of a golf ball hit with initial speed 61 m/s at angle 20$^{\circ }$ by using the ballistic table, Table 3.12 [37]. The value of $m_{\mathrm {b}}/d^{2}$ for a golf ball is 27 kg/m$^{2}$ [37]. Assume $C_{\mathrm {D}} = 0.2$ (because of the very high Re.)

3.107

Football KickingConsider a football kicked 100 feet per second at an angle of 45$^{\circ }$ [57]. At sea level, a spiral kick travels 78 yards with no wind and 65 yards with a headwind of 20 mph. In Denver these corresponding distances are 82 and 70 yards, and the distance is 92 yards with a 20 mph tailwind. A tumbling kick at sea level has a range of 40 yards with no wind, 20 yards with this headwind, and 62 yards with this tailwind. In Denver the tailwind decreases the range from 45 yards to 25 yards. Explain why the order of these ranges makes sense.

3.108

Baseball Spinning ball Throwing Pitching BallsA baseball is rotated by a half revolution by the pitcher during the last 0.04 s before release, due to a constant rotational acceleration due to wrist action and the rotation of the radius on the ulna. Show that the released ball spins at a rate of 1,500 revolutions/min.

Table 3.13 Baseball Spinning ball Throwing Pitching BallsSpeed and spinning of pitches by professional-level pitchers

3.109

Baseball Magnus force Spinning ball Throwing Pitching BallsThe pitcher tries to maximize the rotation speed of the ball ($\omega$, in rad/s) when throwing a curve ball, as seen in Table 3.13. A spinning ball feels the Magnus force (orLift lift) due to this rotation, which changes its path from that of a rotation-free ball. This lift force has magnitude $=\pi \rho r^{3}\omega v/2$ for air mass density $\rho$, ball radius r, spinning rate $\omega$ (in rad/s), and ball speed v. (From [137]):

(a) The Magnus force on a baseball has been determined from measurements to have magnitude (in lb) $=6.4\times 10^{-7}fv$, where f is in revolutions/min and the speed v is in ft/s. Is this consistent with the magnitude of the lift force just given?

(b) How many revolutions do a fast fastball, a slow curve ball, and a medium-speed, medium-rotation-rate knuckleball undergo if each is traveling at a constant forward speed and travels 55 ft before it reaches home plate?

(c)Spinning ball Sketch a diagram showing the initial velocity vector of the ball, how the ball rotates, and the direction of the Magnus forces when a right-handed pitcher rotates the ball in the direction of the screwdriver motion and releases it so that the rotation axis is vertical.

(d)Magnus force If the Magnus force is constant, how much has the ball moved laterally—relative to the expected trajectory with no spinning—when it reaches the plate. (Ignore changes to speed and direction due to drag and gravity, so assume the ball is moving forward and horizontally at 80 mph. Also assume that changes in the trajectory due to the Magnus force are so small that you can assume this trajectory in your calculation.) (To learn more about drag forces see Chap. 7.)

3.110

Baseball Magnus force Spinning ball Throwing Pitching BallsAs in Problem 3.91, a baseball pitcher throws a ball with initial velocity vector component $v_{x,0}$ in the forward direction and $v_{z,0}$ in the vertical direction, but now it has additional constant forces in the vertical z direction $\alpha _{z}g$ and lateral y direction $\alpha _{y}g$ due to its rotation and the Magnus force:

(a) Show that after a time t, its coordinates are: $x=v_{x,0}t$, $y=\alpha _{y}gt^{2}/2$, and $z=v_{z,0}t-(1-\alpha _{z})gt^{2}/2$.

(b) After it travels a distance d in the x direction, show that the ball seems to have fallen a distance $(1-\alpha _{z})gd^{2}/2v_{x,0}^{2}$ and moved laterally a distance $\alpha _{y}gd^{2}/2v_{x,0}^{2}$ from its initial trajectory.

(c) Show that the ball always stays in the tilted plane $z=(v_{z,0}/v_{x,0})t- ((1-\alpha _{z})/\alpha _{y})y$.

(d) Explain why the ball seems to move laterally when $\alpha _{y}\ne 0$, it falls even faster than expected when $\alpha _{z}<0$, and it falls slower than expected—“the rising fastball”—when $\alpha _{z}>0$.

Magnus force(Note that this additional force vector has been assumed to be constant in this simple model. More specifically, the Magnus force has been assumed to be that for a ball thrown with a constant velocity in the x direction. Changes in the force direction and magnitude due to changes in ball velocity due to gravity and the Magnus force itself—and the addition ofDrag drag, make the ball motion even more complex.)

3.111

Baseball Magnus force Spinning ball Throwing Pitching Balls(a) Show that a 80 mph baseball curve ball subjected to a horizontal force of 2.6 oz ($\alpha _{y}=0.5$) curves laterally by 2.1 ft from its expected path (assuming it travels a distance of 60 ft).

(b) In what direction is the spin (given by the right-hand rule) if the ball veers to the right, as seen at the plate. (This is a curve ball as thrown by a right-handed pitcher and a screwball as thrown by a left-handed pitcher.)

(c) How fast is the ball rotating?

3.112

Baseball Magnus force Spinning ball Throwing Pitching BallsCan the Magnus force described in Problem 3.111 explain the “rising fastball”—for which the batter seems to think that the pitched fastball is rising? This could be possible if the ball spun fast enough in the correct direction to give the ball aLift lift force exceeding g. (This corresponds to $\alpha _{z}>1$ in Problem 3.110.) Calculate it for a translational speed of 100 mph and a rotation speed of 2,300 rpm (typical of some excellent pitchers), and show that under the best of circumstances, the Magnus force can account for at most about $\sim$ $2g/3$ (or $\alpha _{z}\sim 2/3$ in Problem 3.110), and so a “rising fastball” is really a consequence of a hitter’s perception and expectation [9, 137].

3.113

Baseball Magnus force Spinning ball Throwing Pitching BallsA right-handed pitcher has similar wrist action when throwing overhand and side arm. When throwing overhand (arm motion in a vertical plane) his ball drops very fast. When throwing side arm (arm motion in a horizontal plane), it moves to the left. When throwing at three-quarter overhand (arm motion in a plane that bisects these two planes), it moves a bit down and to the left. Why?

3.114

Baseball Magnus force Spinning ball Throwing Pitching BallsIn throwing a knuckleball, the pitcher is not concerned with the effects of the Magnus force, but with the sideways force on a ball that is not a perfect sphere because of the stitches on the ball (Fig. 3.85). The pitcher purposely tries to minimize the baseball rotation rate when throwing a knuckleball (Table 3.13), because this force will average to zero for a ball with a moderate rotation rate—and one that is too small for a significant Magnus force. With a slow rotation rate the lateral motion will seem irregular. (This irregular motion also occurs for a ball that has been scuffed—accidentally or on purpose.) The magnitude of thisLift lift force is $C_{\mathrm {L}}A\rho v^{2}/2$, where $C_{\mathrm {L}}$ is the lift coefficient and A is the cross-sectional area, which is similar to (7.63) [137]:

(a) For a knuckle ball, this force is $2.16\times 10^{-5}v^{2}$ (in lb) with v in ft/s. Show that this means that $C_{L}=0.42$.

(b) For a scuffed ball, this force is $1\times 10^{-5}v^{2}$ (with the same units). Show that this means that $C_{L}=0.194$.

(c) Estimate how much a knuckleball can move laterally if it does not rotate at all? (If it does not rotate at all, its motion is predictable, and this is not desirable.)

Fig. 3.85

Boundary layers Baseball Magnus force Spinning ball Throwing Pitching BallsThe flow past a spinless ball can be asymmetric. In this example, the top is smooth and the stitches (or scuff mark) are on the bottom. The seams cause boundary layer turbulence, which delays airstream separation on the bottom surface. The wake moves upward and there is a downward force on the ball (based on [137])

3.115

Cricket Magnus force BallsRedraw Fig. 3.85 for a nonrotating cricket ball (a smooth ball with seams that are stitched at its “equator”). Assume the cricket seam is at a $\sim$45$^{\circ }$ angle with the direction of the flow from the left. Discuss how this seam can affect motion. (Hint: In the rest frame of the ball, the exiting streamlines will not flow along the stitches.)

3.116

Football Spinning ball Throwing BallsWhy does a football thrown spinning about its long axis (a spiral) experience no Magnus force?

3.117

Basketball BallsMost basketball players shoot free throws from the free throw line in an overhand motion, but some professional players, notably Wilt Chamberlain and Rick Barry, have shot them underhanded. In both cases the basketball must go through an 18 in diameter horizontal hoop, with the hoop center 15 in. in front of the backboard. The hoop is 10 ft high on a backboard that is 15 ft from the free throw line. A basketball weighs 21 oz, has a circumference of 30 in, and therefore a diameter of 9.7 in.

Basketball BallsOne study has shown that for a release height of 7 ft, an error in release angle has the minimum affect on accuracy for release angles between 49$^{\circ }$ and $55^{\circ }$, and this corresponds to entry angles between 38$^{\circ }$ and $45^{\circ }$, both relative to the horizontal [66, 77]. More recent analysis suggest that the optimal release angle is between 51$^{\circ }$ and $56^{\circ }$ with release speeds between 20.5 and 24.0 ft/s. SpinSpinning ball due to bendingElbows elbows andKnees knees and snapping the wrist at the moment of release addsLift lifts due to the Magnus force [66].

Basketball BallsAssume the ball goes through the middle of the hoop at a $41^{\circ }$ angle to the horizontal (which is the entry angle). Assume that the 6 ft tall basketball player releases the ball with arms extended vertically for the overhand motion and with outstretched straight arms at shoulder level for the underhand motion. Sketch the motion and determine the initial speed and angle of release for both cases. (Hint: Use the anthropometric data in Chap. 1.)

3.118

Shot puttingA shot putter releases a 7.25 kg shot at a height 2.2 m from the ground. If the shot is released at the angle that maximizes the travel distance (see Problem 3.96), it travels 23.06 m. Find this angle and the initial shot speed [90]. (Ignore drag.)

3.119

Shot putting(a) Find the initial kinetic energy and the maximum increase in the potential energy of the shot in Problem 3.118.

(b) If the putt is 20% efficient, how much energy is used in putting the shot (in J and kcal).

(c) If the putt is pushed from its resting point on the shoulder as the arm is extended, it is accelerated along a distance corresponding to the length of the whole arm, or about 0.7 m. During this acceleration phase (at this extension and release angle), what is the increase in potential energy and how does it compare to the energies in part (a)?

(d) What is the force exerted to accelerate the shot, if this force exerted during this extension phase is constant?

(e)Muscles If the muscles used to accelerate the shot can exert 20 N of force for each cm$^{2}$ of effective cross-sectional area, what is the net cross-sectional area of these muscles? If this were a muscle with a circular cross-section, what would its diameter be?

3.120

Hammer throwingA hammer thrower releases a 7.25 kg hammer at a height 3.5 m from the ground. If the hammer is released at the angle needed so it can travel a maximum distance (see Problem 3.96), it travels 102 m. Find this angle and the initial hammer speed [90]. (Ignore drag.)

3.121

Hammer throwing(a) Find the initial kinetic energy and the maximum increase in the potential energy of the hammer in Problem 3.120.

(b) If the throw is 20% efficient, how much energy is used in throwing the hammer (in J and kcal).

3.122

Bouncing Balls FrictionDerive (3.145)–(3.148) for the flight of a bouncing ball with enough friction so it stopsSkidding ball skidding during the impact, using the supplied relations.

3.123

Bouncing Balls FrictionDerive (3.150)–(3.153) for the flight of a bouncing ball with too little friction to stopSkidding ball skidding during the impact, using the supplied relations.

3.124

Bouncing Balls FrictionShow thatSkidding ball skidding stops during the course of a bounce when $\mu _{\mathrm {k}} v (1+e) \ge \frac{2}{7} (u+r \omega )$, by equating (3.145) and (3.150).

3.125

Bouncing Balls FrictionA solid ball with diameter 70 mm moving at 5 m/s hits a hard surface ($\mu _{\mathrm {k}} = 0.6$, $e= 0.7$) at angle of 30$^{\circ }$ [36].

(a) If the ball is not spinning, show that it has a rebound speed of 3.51 m/s and angle 30$^{\circ }$.

(b)Spinning ball If it is spinning, show that as long as its spinning rate is less than 371 rad/s it will stopSkidding ball skidding before the end of the collision (see Problem 3.124).

(c) Show that as the top spin of the incident ball increases from 100 to 200 to 300 rad/s, its rebound speed increases from 4.11 to 4.84 to 5.66 m/s and rebound angle increases from 42 to 51 to 58$^{\circ }$.

(d) Find the rebound speed and angle for a ball incident with a backspin of 100, 200 and 300 rad/s.

3.126

Skidding ball Friction BallsShow that a solid spherical ball travels a distance $(12/49) v_{0}^{2}/ \mu _{\mathrm {k}} g$ while it is skidding.

3.127

Rolling ball Friction Balls Skidding ballUsing the notation of the chapter, find the final speed of a hollow spherical ball when it starts to roll in terms of its initial speed. (It has a moment of inertia about an axis through its center of mass of $(2/3) m_{\mathrm {b}}r^{2}$.)

3.128

MusclesThe forces exerted by muscles are proportional to their physiological cross-sectional area PCA. For each of the following cases, the result is roughly proportional to (PCA)$^{x}$ (ignoring the force needed to counter the effects of gravity on the body, etc.). Find x in each case. (It should be either 0.5 or 1):

(a)Weightlifting, training The weight lifted by a weightlifter.

(b)Pitching The speed of a ball thrown by a pitcher.

(c)Throwing The farthest a ball can be thrown.

(d)Swinging How fast a bat can be swung. (Is this similar to case (b))?

(e)Hitting balls Striking, hitting, punching The speed of a ball after being hit by a bat. (Assume the pitcher and hitter have muscles proportional to PCA.)

(f) How far a ball can be hit.

Collisions of Humans

3.129

SlippingA professor runs to a waiting elevator and slips on a small drop of water. He lands on hisKnees knees (between the patellas and patella tendons) and his open hands. Explain why this is preferable to landing only on his knees.

3.130

(a) Show that the straight lines denoting specific deceleration times and decelerations in Fig. 3.61 are expected from simple kinematics, for constant deceleration rates. Note that this figure is a log-log plot.

(b) The slopes of the lines for constant deceleration times in this figure seem to be twice those for constant decelerations. Does this makes sense?

3.131

Let us say that as we land at the end of a jump we use the muscles in our knees to help cushion the collision. (Earlier in this chapter we considered a case where we ignored this potential cushioning.) The center of mass of a person of mass $m_{\mathrm {b}}$ falls a distance h before the person’s feet make contact with the ground, and then the person’s center of mass is lowered by a distance s and the reaction force from the ground F is used to cushion the collision:

(a) By using conservation of energy, show that the required force is $F=m_{\mathrm {b}}g(1+h/s)$.

(b)Injuries Sports injuries If we land on both feet and want to keep stresses to no more than 10% of the UCS, show that the maximum force felt by the feet during the collision should be no more than $10^{4}$ N. (Use the analysis in the text for landing with stiff legs.)

(c) If we slow the fall by bending ourKnees knees by 0.5 m, what is the maximum height from which a 70 kg person can safely land? (Do not try this!!!)

3.132

In what ways are the GSI and the impulse during a collision similar and yet different?

3.133

Skating Injuries Sports injuriesA skater falls on ice [13]. Assume that his head hits the ice with a speed corresponding to free fall from a height of 6 ft.

(a) Assuming a constant force during impact, show the GSI for this collision is GSI $=(t_{\mathrm {f}}/\Delta t)^{2.5}\Delta t$, where $t_{\mathrm {f}}$ is the time of the free fall and $\Delta t$ is the collision time.

(b) Calculate the value of the GSI for collision times $\Delta t=1$, 2, 5, and 10 ms.

(c) Let $H=6$ ft be the height of the free fall and $\Delta s$ be the total compressional distance of the collision, which is the flattening experienced by the skin and/or protectivePadding padding. Show that the GSI can be expressed as $\mathrm{GSI} =(H/\Delta s)^{1.5}\sqrt{2H/g}$.

(d) Calculate the GSI for $\Delta s=0.1$, 0.2, 0.5, and 1.0 cm.

(e) Identify the range of values of $\Delta t$ in case (b) and $\Delta s$ in case (d), for which serious injury can be expected.

3.134

InjuriesExplore how the GSI and HIC-15 indices differ for constant deceleration, when:

(a) The magnitude of the deceleration is $\alpha g$ and it lasts for a time $\tau$.

(b) $\alpha =70$ and $\tau =15$ ms.

(c) $\alpha =40$ and $\tau =60$ ms.

(d) What can you say about the weighting of parameters for each index and how each depends on the duration of the collision?

3.135

InjuriesLet us explore how the GSI and HIC indices change when the rate of deceleration is not constant during the collision. Consider an initial speed v and a total collision time $\tau$ ($<$15 ms). It will be simpler if we express v as $\beta g\tau$, where $\beta$ is a constant. The deceleration a(t) is expressed as $\alpha (t)g$. For each of these cases, first confirm that the speed decreases to 0 in a time $\tau$ and plot $\alpha (t)$ on the same graph, and then calculate general expressions for the GSI and HIC for each:

(a) $\alpha (t)=\beta$ is constant during the time $\tau$.

(b) $\alpha (t)=1.5\beta$ from time 0 to $\tau /2$ and $0.5\beta$ from time $\tau /2$ to $\tau$.

(c) $\alpha (t)=2(1-\tau /t)\beta$, so the deceleration decreases linearly from time 0 to $\tau$.

(d) How do the GSI and HIC differ? Which is more sensitive to changes in deceleration?

3.136

Injuries DyingIn the text the GSI was calculated for an elastic and inelastic collisions of a head with $v_{\mathrm {i}}=50$ mph and a collision time of 10 ms. For what collision times would the elastic collision be expected to be fatal and the inelastic collision expected to be survivable?

3.137

Modify (3.116) for a partially elastic collision with a coefficient of restitution e, again with a very massive object.

3.138

Injuries Dying(a) Calculate the GSI for the partially elastic collision of a head moving with $v_{\mathrm {i}}=25$ mph and a collision time of 20 ms with a very massive object, for a general coefficient of restitution e, and then specifically for $e=0$ (totally inelastic collision), 0.5, and 1.0 (totally elastic collision).

(b) Over what ranges of e will the collision definitely be fatal, likely be fatal, likely to not be fatal but likely result in a significant injury, and likely to lead to a relatively minor injury only?

3.139

Baseball Injuries Sports injuries Batting(a) A pitcher throws a baseball at the head of an 80 kg batter (who has an average-sized head and who is not wearing a battingHelmets helmet). It hits his head at a speed of 90 mph at normal incidence and the collision is elastic. Consider the collision of the ball with the head only (ignore the rest of the body) and calculate the GSI assuming that during the collision the baseball decelerates at a constant rate, the baseball deforms its linear dimension (diameter) by 6%, and the head does not deform at all.

(b) What is the fate of the batter?

3.140

Soccer Injuries Sports injuries Throwing Pitching SoccerSome professional soccer players have the same type of loss of cognitive ability as doBoxing boxers, likely due to the repeated heading of soccer balls. What is the GSI when a 82 kg (180 lb) soccer player hits or redirects the ball with his/her head (a header)? Assume that the 430 g ball is moving at 50 mph and hits the head at a normal angle, so itBouncing bounces back on the original path. Say that the collision with the head is elastic and the ball is squeezed by 3 cm in the collision. Treat the collision as with the head (and not the rest of the body).

3.141

Figure 4.74 shows the deceleration and the square wave function approximation to this measured deceleration for a head impacting aHelmets helmet with initial speed 5.63 m/s:

(a) Does this square wave approximation stop the head in the time shown?

(b)Injuries Calculate the GSI and HIC-15 for the square wave function approximation.

3.142

Football Injuries Sports injuriesIn professional footballFootball, a linebacker and fullback, both weighing 245 lb, are racing toward each otherRunning running at 30 ft/s. (Actually, the linebacker is running to the fullback and the fullback is trying to run away from him.) They collide and then both decelerate at a constant rate and become stationary in 0.2 s:

(a) What is the deceleration of each? (Also express your answer in g.)

(b) What is the force on each during the collision?

3.143

Football Injuries Sports injuries Running(a) A big defensive lineman in professional football weighing 310 lb runs at a speed of 24 ft/s into a small quarterback weighing 189 lb, who is initially still. Using conservation of linear momentum, what are their speeds after this collision and in what directions are they moving? Assume all frictional forces during the collision can be neglected, so kinetic energy is also conserved during the collision.

(b) Alternatively, assume that the lineman holds on to the quarterback during and after the collision, i.e., “tackles” or “sacks” him. In this case, what fraction of the initial kinetic energy is lost? If the forces with the ground can be neglected, where did this lost kinetic energy go?

3.144

Injuries Sports injuries Baseball SlidingA base runner has an initial speed of 9 m/s and slides to a stop over 4 m. WhatFriction Coefficient of friction is the coefficient of friction during the slide?

3.145

Injuries Sports injuries Baseball Sliding(a) When you slide into a base at speed $v_{\mathrm {slide,final}}$ your foot stops after crushing into the base a distance d. Find this constant deceleration and, assuming the decelerating force is transmitted to the whole body, this collision force $F_{\mathrm {coll}}$.

(b) Assume this force causes aFractures fracture when this force $F_{\mathrm {coll}}$ is distributed over a bone area A and this force per unit area $F_{\mathrm {coll}}/A$ exceeds the UCS of the bone. Find the threshold sliding speed $v_{\mathrm {slide,fracture}}$ for fracture.

(c) For reasonable values of A and the ultimate compressive stress (UCS) (see Chap. 4), find $v_{\mathrm {slide,fracture}}$.

3.146

Karate(a) Show that the minimum kinetic energy an object of mass M must have to impart a kinetic energy $\mathrm{KE}_{\mathrm {transferred}}$ to a stationary mass of mass m during a collision is $\mathrm{KE}_{\mathrm {initial}} =((m+M)/M) \mathrm{KE}_{\mathrm {transferred}}$ assuming an inelastic collision and $\mathrm{KE}_{\mathrm {initial}} =((m+M) ^{2}/4mM) \mathrm{KE}_{\mathrm {transferred}}$ assuming an elastic collision. These results can be used to model theFractures breaking of boards in karate [144]. We assume the hand (as a fist) with mass $M= 0.7$ kg that strikes the board is theEffective mass effective mass of the body in this collision.

(b) Assume the amount of energy needed to break a given wooden board is 5.3 J (with mass $m = 0.14$ kg) and a given concrete board is 1.6 J (3.2 kg). Show that the model assuming elastic collisions predicts 9.5 J is needed (corresponding to a strike speed of 5.2 m/s) to break the wood board and 2.7 J (2.8 m/s) is needed to break the concrete board, and the model assuming inelastic collisions predicts 6.4 J (4.3 m/s) is needed to break the wood board and 8.9 J (5.0 m/s) is needed to break the concrete board.

(c) Measurements show that the wood board breaks with an initial kinetic energy of 12.3 J (6.1 m/s) and the concrete board with an initial kinetic energy of of 37.1 J (10.6 m/s), so it is easier toFractures break this wood board than this concrete board in practice. What can you conclude about the model predictions, specifically regarding which model agrees the observation that the wood board breaks easier and the absolute numerical predictions themselves?

3.147

Karate FracturesRepeat Problem 3.146 now assuming the combination of the hand and forearm is theEffective mass effective mass that strikes the boards, so $M = 2$ kg [144].

3.148

KarateWhy would it not be advisable to use oak boards in karate demonstrations of breaking boards? (see Table 4.1).

3.149

Diseases and disordersA movie shows one man shooting another. The shooter is firm and steady when firing the gun and the other man recoils greatly when the bullet hits him. Does this make sense?

3.150

Diseases and disordersA bullet of mass 18 g traveling at 400 m/s hits someone’s head and comes to rest in the middle of it. Treat the head as a uniform sphere of radius 8.5 cm and mass 5 kg. Assume that the head undergoes no trauma from the bullet and does not change its shape, and that it can be treated as separate from the rest of the body, and so it is theEffective mass effective mass of the target in this collision.

(a) Find the speed of the head after the bullet stops.

(b) Assuming the bullet decelerates at a constant rate, find the collision time.

(c) Find the GSI for the head due to this collision.

(d) Ignoring the obvious physical damage done by the bullet, what is the fate of the person due to this collision?

3.151

Diseases and disordersUsing the conditions of Problem 3.150 for a bullet lodging in a head, find

(a) the kinetic energy of the bullet before the collision,

(b) the thermal energy deposited in the head by the bullet (which is the difference of total kinetic energies before and after this inelastic collision, assuming no other effects),

(c) the fraction of the initial bullet kinetic energy this represents, and

(d) the temperature rise in the head right after the bullet comes to rest (assuming that the head remains in tact, the specific heat of the head is the same as for the body, that the temperature in the head is always constant, and there is no heat flow from the head).

3.152

Diseases and disordersWhen a bullet hits a head, the head is expected to recoil in the direction of the bullet, away from the shooter. Let us consider the gruesome situation when the bullet goes through the brain and exits it, and a jet of blood and brains follows the bullet out of the brain [142]. The brain (or head), bullet and jet can be treated as three objects. Assume the jet has 10% of the initial kinetic energy of the bullet and that it has a mass $15\times$ that of the bullet. Show that if these circumstances could be modeled in this manner, the head would be expected to recoil in the direction opposite to the flight of the bullet, toward the shooter.

Hitting and Kicking Balls

3.153

Pool, billiardsShow that after a (point-like) object collides elastically with an identical object that is initially at rest, the first one is stationary and the second one moves with the same velocity as the first one had before the collision.

3.154

Baseball Balls Striking, hitting, punching BattingIn terms of the coefficient of restitution e, find the fraction of kinetic energy that is recovered in the collision when a ball falls vertically and bounces from a horizontal floor.

3.155

Basketball BouncingA “super ball,” basketball, volleyball, and softball are dropped from a height of 1.83 m (6 ft) onto a hardwood floor. How high do they bounce?

3.156

Baseball Bouncing BallsA baseball is found to bounce to a height of 0.46, 0.51, and 0.55 m from a height of 1.83 m (6 ft), when it has been previously cooled for 1 h in a freezer, left at room temperature, and heated for 15 min at $225\,^{\circ }$C, respectively. What are the coefficients of restitution for the balls in the three cases? [66]

3.157

Baseball Balls Striking, hitting, punching BattingHow far can a batted baseball travel? Assume the ball speed and bat speed are both 90 mph, $e=0.46$, and the bat is much heavier than the ball. (Go into the rest frame of the bat, analyze the collision, and then return back.) (Your answer will be an overestimate becauseDrag drag is omitted.)

3.158

Baseball Balls Throwing Pitching Balls Batting Striking, hitting, punching SwingingIf the same torque is generated by a pitcher in throwing a ball and a batter in hitting a ball, would the ability to throw a 90 mph fastball mean you could swing a bat with a 60 mph speed of the center of mass of the bat? Assume the moment of inertia for throwing the ball is that of a straight arm about the shoulder and that for hitting a ball is the body with two outstretched arms and a fully extended bat (as in Fig. 3.68).

3.159

Baseball Balls Striking, hitting, punching Batting SwingingDerive (3.127) by using the method described preceding it in the text.

3.160

Baseball Balls Striking, hitting, punching Batting SwingingUse our simple torque model for throwing a ball to show how you would expect the final bat speed to depend on bat weight. Is the dependence used in the text, (3.128), reasonable in terms of its linearity and the magnitude of its coefficients?

3.161

Baseball Striking, hitting, punching Swinging BattingDerive (3.129) by using the method described preceding it in the text.

3.162

Baseball Balls Striking, hitting, punching Swinging BattingUse (3.126)–(3.130) to find the range of bat weights for which the batted ball speed for this example is within 2 mph of the maximum speed (obtained with the optimized bat weight).

3.163

Baseball Balls Striking, hitting, punching SwingingFor the example of optimizing the bat weight for a ball with speed $-80$ mph and a bat speed that decreases with bat weight, find the bat speed before and after the collision with the baseball.

3.164

Baseball BallsUse (3.129 Batting Striking, hitting, punching Swinging) to determine the optimal bat weight and batted ball speed for a major leaguer whose bat speed varies as $v_{\mathrm {bat}}$ (in mph) $=48-0.34W_{\mathrm {bat}}$ (with $W_{\mathrm {bat}}$ in oz). Over what range of bat weights is the batted ball speed within 5% of this maximum speed?

3.165

Baseball Balls Striking, hitting, punching Batting SwingingEquation (3.126) gives the speed of the baseball after it collides with a bat at its center of mass. In this problem we show that the analysis is a bit different if it hits the bat elsewhere, as in Fig. 3.68. We will use notation similar to that in the chapter, so $v_{\mathrm {ball}}^{\prime }$ is the speed of the ball after the collision and $v_{\mathrm {bat}}^{\prime }$ is the speed of the bat center of mass after the collision. (Before the collision these variables are unprimed):

(a) Conservation of linear momentum, (3.108), still holds. Show this is now

$$\begin{aligned} m_{\mathrm {ball}}v_{\mathrm {ball}}+m_{\mathrm {bat}}v_{\mathrm {bat}} =m_{\mathrm {ball}}v_{\mathrm {ball}}^{\prime }+m_{\mathrm {bat}}v_{\mathrm {bat}}^{\prime }. \end{aligned}$$

(3.165)

(b) The total angular momentum of the bat and ball is conserved in the collision. The rotational kinetic energy of the ball is $\simeq$5–10% of its translational kinetic energy (show this), so we can neglect the rotation of the ball. Show that conservation of angular momentum gives

$$\begin{aligned} I_{\mathrm {bat}}(\omega _{\mathrm {bat}}^{\prime }-\omega _{\mathrm {bat}}) +Bm_{\mathrm {ball}}(v_{\mathrm {ball}}^{\prime }-v_{\mathrm {ball}})=0, \end{aligned}$$

(3.166)

where $I_{\mathrm {bat}}$ is the moment of inertia of the bat about its center of mass, and $\omega _{\mathrm {bat}}$ and $\omega _{\mathrm {bat}}^{\prime }$ are the angular velocity of the bat before and after the collision, respectively.

(c) The coefficient of restitution equation, (3.109), now involves the speed of the bat at the point of impact. Show that it now becomes

$$\begin{aligned} e=- \frac{v_{\mathrm {ball}}^{\prime }-v_{\mathrm {bat}}-B \omega _{\mathrm {bat}}^{\prime }}{v_{\mathrm {ball}}-v_{\mathrm {bat}}-B\omega _{\mathrm {bat}}}. \end{aligned}$$

(3.167)

(d)Baseball Balls Batting Striking, hitting, punching Swinging Solve these three equations to show that the speed of the ball after the collision is

$$\begin{aligned} v_{\mathrm {ball}}^{\prime }=\frac{(m_{\mathrm {ball}}-em_{\mathrm {bat}} +m_{\mathrm {ball}}m_{\mathrm {bat}}B^{2}/I_{\mathrm {bat}})v_{\mathrm {ball}}+m_{\mathrm {bat}}(1+e) (v_{\mathrm {bat}}+B\omega _{\mathrm {bat}})}{m_{\mathrm {\mathrm {ball}}}+m_{\mathrm {bat}} + m_{\mathrm {ball}}m_{\mathrm {bat}}B^{2}/I_{\mathrm {bat}}} \end{aligned}$$

(3.168)

(e) Show that this reduces to (3.126) when the ball hits the bat center of mass ($B=0$).

For more details see [17, 18, 137].

3.166

Batting Baseball Throwing Pitching Balls Striking, hitting, punching SwingingA ball is pitched at 85 mph and the batter retracts the bat to bunt the ball. How fast does he have to retract it (in mph) so the ball rebounds with a speed of 5 mph? (Assume all motion is in one dimension. Ignore the redirection of the ball to the ground by the bat and any rotation of the pitched ball.)

3.167

Batting Baseball Striking, hitting, punching SwingingA force F is applied to a bat (such as in a collision with a ball), from the left to the point labeled as the sweet spot or center of percussion in Fig. 3.69a [36]. Let us examine the motion at a point x away (the bat handle in the figure), which we will call the pivot. The center of mass of the bat (as in the figure) is y away from the pivot. The bat has a mass m and moment of inertia I about its center of mass. We assume, without losing generality, that the force is constant over a time t.

(a) Show that the effect of the force on the center of mass is to move the bat to the right (in the figure) with a linear speed Ft / m.

(b) Show that the effect of the torque is to cause a rotation about the center of mass with a rotational speed $\omega = F(x-y)t/I$, and that this leads to a rotational speed of the bat handle of $F(x-y)yt/I$ in the direction shown in the figure (to the left).

(c) Explain why there is no motion of the pivot (the bat handle) when the bat is struck so that $x= y + I/my$ for this collision at the center of percussion.

3.168

Batting Striking, hitting, punching Swinging BaseballA bat of mass m and length L has a uniform cross section, so it has a moment of inertia of $mL^{2}/12$ about its center of mass. Use the results of Problem 3.167 to show that the center of percussion is a distance 2L / 3 from the bat handle, which is considered to be one of the ends of the bat, or 2/3 of the way from the bat handle [36].

3.169

Pool, billiardsA pool ball of mass $m_{\mathrm {b}}$, radius r, and moment of inertia $(2/5) m_{\mathrm {b}}r^{2}$ is hit with a cue that provides a horizontal force at a height x above the table. Show that when the ball is hit at a height $x = (7/10)(2r)$, which is 70% of the ball diameter, the ballRolling ball rolls (withSpinning ball topspin) without any initialSkidding ball skidding. (Hint: One way of solving this problem is to show that this requirement is formally equivalent to hitting the object at its center of percussion, as in Problem 3.167 and Fig. 3.69a, with the point of contact between the ball and table identified as the pivot [36].)

3.170

Basketball Balls BouncingA slow-footed basketball player who likes to shoot off the backboard wants to deflate a well-inflated basketball, but his fast-footed point guard teammate (who dribbles (bounces) the ball) objects to this. Why?

3.171

Football Balls KickingWhy do football punters and placekickers want to kick balls that are very well inflated?

3.172

Football Balls KickingKicking a football, such as a “place kick,” is a form of a collision – of the foot with the football:

(a) If the average force on the ball is 450 lb during the kick and the kick lasts for 8 ms, how fast does the ball move after the kick? Assume the football weighs 0.91 lb.

(b) How far does the ball travel (in yards) if its takeoff angle is 45$^{\circ }$ and drag is neglected?

3.173

Football Kicking BallsA football player kicks a football at a takeoff angle of 45$^{\circ }$; it lands 50 yd away (ignoring air resistance). Assume the person has a mass of 80 kg and height of 1.9 m (use Tables 1.6 and 1.7, and Fig. 1.15), and the mass of the football can be ignored during the kick. Also, assume that at the end of the kick, the kicker’s foot is moving at the same speed as the football. Examine the kicking leg by itself, so it is theEffective mass effective mass of the body in this collision, assuming that it has a constant mass per unit length, and find the average power generated by the leg during the kick in W and in hp if the duration of the kick is 0.2 s. If the body can produce this mechanical motion with 15% efficiency, what average power does it need, in W and in hp, to achieve this motion?

3.174

Soccer Kicking BallsAssume that after a soccer ball is kicked, the ball moves with the same speed as the player’s foot. For a soccer player with the dimensions of the average human in Chap. 1, how fast does the soccer ball move with thigh and shank (lower leg) motion as in Fig. 3.86?

Fig. 3.86

The angular speed of theHip hip joint (thigh) andKnees knee (shank) duringKicking kicking of aSoccer soccer ball (The unusual units of 1/66 s are from the 66/s frame rate of the photographs taken by [118].) (based on [97, 118]). For Problem 3.174

3.175

Soccer Kicking BallsDoes Problem 3.174 assume that the collision is inelastic, partially elastic, or elastic? Is this a good assumption? If not, how would you correct the approach in that problem?

3.176

Striking, hitting, punching Baseball BallsExplain why aMagnus force Spinning ball baseball hit to the right side of the park develops spin that makes it curve to the right and one hit to the left side develops spin that makes it curve left.

3.177

Tennis Balls Ping pong, table tennis Spinning ballUsing a racket to impart topspin and backspin is very important in ping pong and tennis. Explain why topspin is imparted in the collision depicted in Fig. 3.70a and backspin in Fig. 3.70b, and describe the flight of the ball [60].

3.178

Golf SwingingThe club head of the driver in a golf drive moves in a circular arc with diameter equal to the shaft length 1 m with a speed 45 m/s. Find the centrifugal pull on the hands if the gravitational pull of the driver head is 2 N. (You will be able to understand why you need to grip the club strongly during the drive by comparing this result with the gravitational pull [36].)