Kinematics: straight-line motion using calculus

Learning objectives

Relate displacement, velocity and acceleration by differentiation, v = dx/dt and a = dv/dt = d²x/dt², and by integration in the reverse direction

Find the displacement or velocity at a given time from a velocity or acceleration function together with an initial condition, determining the constant of integration correctly

Determine when a particle is instantaneously at rest by solving v = 0, when it changes direction, and when its acceleration is zero so that the velocity is greatest or least

Distinguish total distance travelled from displacement by splitting the motion at the times when the velocity changes sign and adding the magnitudes of the separate parts

Interpret displacement–time and velocity–time graphs, reading the velocity as the gradient of a displacement–time graph and the displacement as the area under a velocity–time graph

Solve problems involving motion with variable acceleration, including finding the maximum speed and the time taken to return to the starting point