Systems of Point Masses; Collisions

Appendices

Summary

• The centre of mass of a system of N point masses m _i with the position vectors $\boldsymbol{r}_{i}$ has the position vector

  $$\displaystyle\boldsymbol{r}_{\mathrm{S}}=\frac{1}{\sum m_{i}}\sum m_{i}\boldsymbol{r}_{i}=\frac{1}{M}\sum m_{i}\boldsymbol{r}_{i}{\;}.$$

• The coordinate system with the CM as origin is called the centre-of-mass system

• The vector sum of all momenta $m_{i}\boldsymbol{v}_{i}$ of the masses m _i in the CM-system is always zero.

• The reduced mass μ of two masses m ₁ and m ₂ is defined as

  $$\displaystyle\mu=\frac{m_{1}\cdot m_{2}}{m_{1}+m_{2}}{\;}.$$

• The relative motion of two particles with the mutual interaction forces $\boldsymbol{F}_{12}=\boldsymbol{-F}_{21}$ can be reduced to the motion of a single particle with the reduced mass μ which moves with the velocity $\boldsymbol{v}_{12}=\boldsymbol{v}_{1}-\boldsymbol{v}_{2}$ around the centre of m ₁.

• A system of particles with masses m _i , where no external forces are present is called a closed system. The total momentum and the total angular momentum of a closed system are always constant, i. e. they do not change with time (conservation laws for momentum and angular momentum).

• In elastic collisions between two particles the total kinetic energy and the total momentum are conserved. For inelastic collisions the total momentum is also conserved but part of the initial kinetic energy is transferred into internal energy (e. g. potential energy or kinetic energy of the building blocks of composed collision partners). Inelastic collisions can only occur if at least one of the collision partners has a substructure, i. e. is compound of smaller entities.

• While for elastic collisions in the lab-system the kinetic energies E _i of the individual partners change (although the total energy is conserved), in the CM-system also the E _i are conserved.

• In inelastic collisions only the kinetic energy $\tfrac{1}{2}\mu v_{12}^{2}$ of the relative motion can be transferred into internal energy. At least the part $\tfrac{1}{2}Mv_{\mathrm{S}}^{2}$ of the CM-motion must be preserved as kinetic energy of the collision partners.

• The collision between two particles with masses m ₁ and m ₂ can be reduced in the CM-system to the scattering of a single particle with reduced mass

  $$\displaystyle\mu=\frac{m_{1}\cdot m_{2}}{m_{1}+m_{2}}$$

  by a particle with mass $m\infty$ fixed in the CM. This can be also described by the scattering in a potential depending on the interaction force between the two particles.

• The deflection angle $\varphi$ of the particle in the CM-system depends on the impact parameter b, the reduced mass μ, the initial kinetic energy $\tfrac{1}{2}\mu v_{0}^{2}$ and the radial dependence of the interaction potential.

• The evaluation of collisions at relativistic velocities v demands the consideration of the relativistic mass increase. Then also energy and momentum conservation remain valid.

• The conservation laws for energy, momentum and angular momentum can be ascribed to general symmetry principles, as the homogeneity of space and time and the isotropy of space.

Problems

4.1

Two particles with masses $m_{1}=m$ and $m_{2}=3m$ suffer a central collision. What are their velocities $v_{1}^{\prime}$ and $v_{2}^{\prime}$ after the collision if the two particles had equal but opposite velocities $v_{1}=-v_{2}$ before the collision

• a) For a completely elastic collision

• b) For a completely inelastic collision?

4.2

A wooden block with mass $m_{1}=1\,\mathrm{k}\mathrm{g}$ hangs on a wire with length $L=1\,\mathrm{m}$. A bullet with mass $m_{2}=20\,\mathrm{g}$ is shot with the velocity $v=10^{3}\,\mathrm{m}/\mathrm{s}$ into the block and sticks there. What is the maximum deflection angle of the block?

4.3

A proton with the velocity v ₁ collides elastically with a deuteron (nucleus consisting of proton and neutron) at rest. After the collision the deuteron flies under the angle of $45^{\circ}$ against v ₁. Determine

• a) the deflection angle $\theta_{1}$ of the proton

• b) the CM-velocity

• c) the velocities $v_{1}^{\prime}$ and $v_{2}^{\prime}$ of proton and deuteron after the collision.

4.4

A particle with mass $m_{1}=2\,\mathrm{k}\mathrm{g}$ has the velocity $\boldsymbol{v}_{1}=\{3\hat{e}_{x}+2\hat{e}_{y}-\hat{e}_{z}\}\,\mathrm{m}/\mathrm{s}$. It collides completely inelastic with a particle of mass $m_{2}=3\,\mathrm{k}\mathrm{g}$ and velocity $\boldsymbol{v}_{2}=\{-2\hat{e}_{x}+2\hat{e}_{y}+4\hat{e}_{z}\}$. Determine

• a) The kinetic energies of the two particle before the collision in the lab-system and the CM-system.

• b) Velocity and kinetic energy of the compound particle after the collision.

• c) Which fraction of the initial kinetic energy has been converted into internal energy? Calculate this fraction in the lab-system and the CM-system.

4.5

A mass $m_{1}=1\,\mathrm{k}\mathrm{g}$ with a velocity $v_{1}=4\,\mathrm{m}/\mathrm{s}$ collides with a mass $m_{2}=2\,\mathrm{k}\mathrm{g}$. After the collision m ₁ moves with $v_{1}^{\prime}=\sqrt{8\,\mathrm{m}/\mathrm{s}}$ under an angle of $45^{\circ}$ against $\boldsymbol{v}_{1}$ and m ₂ with $v_{2}^{\prime}=\sqrt{2\,\mathrm{m}/\mathrm{s}}$ under an angle of $-45^{\circ}$

• a) What was the velocity v ₂?

• b) Which fraction of the initial kinetic energy has been converted into internal energy in the lab-system and the CM-system?

• c) How large are the deflection angles $\vartheta_{1}$ and $\vartheta_{2}$ in the CM-system?

4.6

Two cuboids with masses $m_{1}=1\,\mathrm{k}\mathrm{g}$ and $m_{2}<m_{1}$ slide frictionless on an air-track, which is blocked on both sides by a vertical barrier (Fig. 4.33). Initially m ₁ is at rest and m ₂ moves with constant velocity $v_{2}=0.5\,\mathrm{m}/\mathrm{s}$ to the left. After the collision with m ₁ the mass m ₂ is reflected to the right, collides with the barrier $(m=\infty)$ and slides again to the left. We assume that all collisions are completely elastic.

• a) What is the ratio $m_{1}/m_{2}$ if the two masses finally move to the left with equal velocities?

• b) How large should m ₂ be in order to catch m ₁ before it reaches the left barrier?

• c) Where collide the two masses at the second collision for $m_{2}=0.5\,\mathrm{k}\mathrm{g}$?

Fig. 4.33

Illustrating Probl. 4.6

4.7

A steel ball with mass $m_{1}=1\,\mathrm{k}\mathrm{g}$ hangs on a wire with $L=1\,\mathrm{m}$, vertically above the left edge of a resting mass $m_{2}=5\,\mathrm{k}\mathrm{g}$ which can slide without friction on a horizontal air-track. (Fig. 4.34). The steel ball with the wire is lifted by an angle $\varphi=90^{\circ}$ from the vertical into the horizontal position and then released. It collides elastically with the glider. What is the maximum angle $\varphi$ of m ₁ after the collision?

Fig. 4.34

Illustration of Probl. 4.7

4.8

An elevator ascends with constant velocity $v=2\,\mathrm{m}/\mathrm{s}$. When its ceiling is still $30\,\mathrm{m}$ below the upper point A of the lift shaft a ball is released from A which falls freely down and hits elastically the ceiling of the elevator, from where it is elastically reflected upwards.

• a) Where does it hit the elevator ceiling?

• b) What is its maximum height after the reflection?

• c) Where does it hit the elevator ceiling a second time?

4.9

An α-particle (nucleus of the He-atom) hits with the velocity $\boldsymbol{v}_{1}$ an oxygen nucleus at rest $(m_{2}=4m_{1})$. The α-particle is deflected by $64^{\circ}$, the oxygen nucleus by $-51^{\circ}$ against $\boldsymbol{v}_{1}$. The collision is completely elastic.

• a) What is the ratio $v_{1}^{\prime}/v_{2}^{\prime}$ of the velocities after the collision?

• b) What is the ratio of the kinetic energies after the collision?

4.10

A particle has in a system S a kinetic energy of $6\,\mathrm{G}\mathrm{eV}$. and the momentum $P=6\,\mathrm{G}\mathrm{eV}/c$. What is its energy in a system $S^{\prime}$, where its momentum is measured as $5\,\mathrm{G}\mathrm{eV}/c$? What is the relative velocity of $S^{\prime}$ against S?