Abstract
This chapter introduces kinematics through an exposition that combines analytical and numerical mathematical techniques. The description of motion, including aspects such as velocity and acceleration, is explained starting from a numerical example—Usain Bolt’s race in the Beijing Olympics in 2008. Then we introduce the basic methods to determine the motion of an object given its acceleration—through analytical integration, the solution of differential equations analytically and numerically. Worked examples combining numerical and analytical exposition as well as realistic data, introduces the student to robust problem-solving methods. Problems and projects that combine analytical and numerical methods are provided.
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The attentive reader may realize that the average acceleration should really be defined in terms of the change in instantaneous velocity: \(\bar{a} = (v(t + \varDelta t) - v(t) )/\varDelta t\) and not in terms of the average velocity as done here. However, this small difference in definitions becomes insignificant when the time interval becomes sufficiently small.
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Notice that we need two initial conditions, \(v(t_0) = v_0\) and \(x(t_0) = x_0\) to determine the position from the acceleration: This is because we need to first calculate the velocities, and this requires an initial condition of the velocities, and then calculate the position, and this also requires an initial condition, this time on the position.
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© 2015 Springer International Publishing Switzerland
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Malthe-Sorenssen, A. (2015). Motion in One Dimension. In: Elementary Mechanics Using Matlab. Undergraduate Lecture Notes in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-19587-2_4
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DOI: https://doi.org/10.1007/978-3-319-19587-2_4
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Publisher Name: Springer, Cham
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Online ISBN: 978-3-319-19587-2
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