Differentiating motion

Objective. Obtain velocity and acceleration from displacement.

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Learn: the key idea

Differentiating motion means obtaining velocity and acceleration from a displacement function s(t) by applying calculus: velocity is the rate of change of displacement, v = ds/dt, and acceleration is the rate of change of velocity, a = dv/dt = d²s/dt². This applies even when displacement is a complicated or non-polynomial function of time, unlike suvat equations which require constant acceleration. For example, if s = t³ − 3t² + 2t, then v = 3t² − 6t + 2, and a = 6t − 6, giving instantaneous values at any time t.

Worked example

A particle has displacement s = 2t³ − 9t² + 12t metres. Find the velocity and acceleration of the particle at t = 2 seconds.

1 $v = ds/dt = 6t² - 18t + 12$
2 $At t = 2: v = 6(4) - 18(2) + 12 = 24 - 36 + 12 = 0$
3 $a = dv/dt = 12t - 18$
4 $At t = 2: a = 12(2) - 18 = 24 - 18 = 6$

Answer: v = 0 m/s, a = 6 m/s² at t = 2

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Common mistake. Students apply suvat formulas to a displacement given as a general function of time, which is only valid for constant acceleration. Use differentiation whenever displacement is given as a non-linear function of t beyond a simple quadratic.

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