Kinematics

Maths: Analysis and Approaches · Calculus. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

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The idea

Kinematics describes motion along a straight line. Displacement s is the position measured from a fixed point and can be negative. Velocity v is the rate of change of displacement, and acceleration a is the rate of change of velocity. So you differentiate to go from s to v to a, and integrate to go back, using the starting values to find each constant.

A particle is at rest when v = 0. It changes direction only if v changes sign there.

Distance is not the same as displacement. Displacement is the change in position. Total distance counts every metre travelled, forwards and backwards, so you must split the motion at each time the particle turns round.

Rules to remember

Common mistake

Giving the displacement when the question asks for total distance. If the particle turns round, find the distance for each part of the motion and add the sizes.

Worked example

A particle has velocity v = 6t − 3t2 m s−1 for 0 ≤ t ≤ 3. Find the total distance travelled.

  1. v = 3t(2 − t) = 0 at t = 0 and t = 2, and v changes sign at t = 2
  2. ∫v dt = 3t2 − t3. From 0 to 2: (12 − 8) − 0 = 4 m
  3. From 2 to 3: (27 − 27) − (12 − 8) = −4, so 4 m backwards
  4. Total distance = 4 + 4 = 8 m (the displacement is 0 m)

Answer: 8 m

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