Motion Along a Straight Line
Kinematics of linear motion applies differentiation and integration to displacement, velocity and acceleration along a straight line. SPM tests the relationships between s, v and a, the rest condition, maximum or minimum displacement, and total distance travelled.
Core rules
v = ds/dt and a = dv/dt = d²s/dt². A particle is at rest when v = 0, and its displacement is stationary there. It moves forwards when v > 0 and backwards when v < 0, and speeds up when v and a have the same sign. Total distance travelled is ∫|v| dt, split at the times when v = 0; displacement is ∫v dt.
Worked example
s = t³ − 6t² + 9t. Then v = 3t² − 12t + 9 = 3(t−1)(t−3), so the particle is at rest at t = 1 and t = 3. At t = 0, s = 0; at t = 1, s = 4; at t = 3, s = 0. The particle moves forwards to s = 4 and then backwards to s = 0, so the total distance is 4 + 4 = 8 units even though the displacement is 0.
Traps
Reporting displacement when distance is asked; forgetting to split the integral at v = 0; assuming a = 0 means the particle is at rest; sign errors when differentiating.