Quick questions on Kinematics: suvat equations, motion graphs and calculus of motion - Edexcel A-Level Maths

On a displacement-time graph the gradient gives velocity. On a velocity-time graph the gradient gives acceleration and the area under the graph gives displacement. These let you read motion straight from a sketch. A horizontal line on a displacement-time graph means the body is stationary, while a horizontal line on a velocity-time graph means constant velocity.

When acceleration varies, use calculus. If displacement is $s(t)$, then velocity is $v = \dfrac{ds}{dt}$ and acceleration is $a = \dfrac{dv}{dt} = \dfrac{d^2s}{dt^2}$. Reversing, $s = \int v\,dt$ and $v = \int a\,dt$, with constants found from initial conditions. A turning point of the displacement happens where $v = 0$, and a maximum or minimum velocity happens where $a = 0$, so calculus also locates the moments when the body is instantaneously at rest or moving fastest.

Choose a positive direction and keep displacement, velocity and acceleration consistent with it.

A ball is thrown up at $20$ m per second. Taking $g = 9.8$, find the time to reach its highest point. [3 marks]

A particle has velocity $v = 3t^2$ m per second. Find its acceleration at $t = 2$. [2 marks]