A-level · Kinematics

Kinematics with calculus (a → v → s)Exam-style questions

Why integrating acceleration gives velocity and integrating velocity gives displacement, where the suvat formulas actually come from, and why they silently break the moment acceleration varies.

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22 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 2 Level 3 Level 4 Level 5

1
In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable.

A particle $P$ travels along a straight line. Its speed at time $t$ seconds is $v$ m s $^{-1}$ , where

$v = \begin{cases} 2t + 3 & 0 \leqslant t \leqslant k \\ 3\sqrt{t} + 12 & k < t \leqslant 30 \end{cases}$

and $k$ is a constant.

Given that $v$ is continuous for $0 \leqslant t \leqslant 30$ ,
(a) show that $k = 9$ . You must make your reasoning clear.

(3 marks)
(b) Find the acceleration of $P$ when $t = 16$

(3 marks)

When $t = 0$ , $P$ is at the fixed point $O$ on the line.
(c) Find the distance $OP$ when $t = 25$

(7 marks)
Level 5 · 13 marksTry it
2
In this question, $\mathbf{i}$ and $\mathbf{j}$ denote unit vectors due east and due north respectively.

A jet ski $J$ moves on the surface of a calm sea. $J$ , together with its rider, has mass $230$ kg and is modelled as a particle. At time $t$ seconds, where $t \geq 0$ , the velocity of $J$ is $\mathbf{v}\ \text{m s}^{-1}$ and the resultant horizontal force acting on $J$ is $\mathbf{F}$ N, where

$\mathbf{v} = (4t - 11)\mathbf{i} + (t^2 - 5t + 9)\mathbf{j}.$
(a) Show that $J$ is never stationary.

(2 marks)
(b) Find, in terms of $\mathbf{i}$ and $\mathbf{j}$ , the acceleration of $J$ at time $t$ .

(1 mark)
(c) Determine the magnitude of $\mathbf{F}$ at the instant when $J$ is moving north-west. Give your answer correct to 3 significant figures.

(5 marks)

The position vector of $J$ relative to a fixed point $O$ is $\mathbf{r}$ m. When $t = 3$ , $\mathbf{r} = -29\mathbf{i} - 7.5\mathbf{j}$ .
(d) Determine the bearing of $J$ from $O$ when $t = 6$ .

(5 marks)
Level 5 · 13 marksTry it
3
In this question, $\mathbf{i}$ and $\mathbf{j}$ denote unit vectors due east and due north respectively.

Two horizontal forces act on a particle $P$ of mass $5$ kg, which moves on a smooth horizontal plane. One force is constant and equal to $(-13\mathbf{i} + 12\mathbf{j})$ N; at time $t$ seconds, where $t \geq 0$ , the other force is $\left((3t^2 + 1)\mathbf{i} - 6t\mathbf{j}\right)$ N. No other horizontal force acts on $P$ . When $t = 0$ , $P$ is at rest at the fixed point $O$ .
(a) Determine the value of $t$ for which the forces acting on $P$ are in equilibrium.

(2 marks)
(b) Determine the speed of $P$ at the instant when $P$ is moving due east.

(6 marks)
(c) Determine the distance of $P$ from $O$ when $t = 5$ .

(5 marks)
Level 5 · 13 marksTry it
4
A tram starts from rest at a stop and moves along a straight horizontal track towards the next stop. For the first 40 seconds of its journey, its velocity $v\text{ m s}^{-1}$ at time $t$ seconds after it starts is modelled by

$v = kt - 0.015t^2,$

where $k$ is a constant.

At the instant when $t = 40$ , the tram is decelerating at $0.3\text{ m s}^{-2}$ . After this, the tram continues to decelerate at a constant $0.3\text{ m s}^{-2}$ until it comes to rest at the next stop.
(a) Find the value of $k$ .

(3 marks)
(b) Find the distance between the two stops.

(7 marks)
Level 5 · 10 marksTry it
5
The velocity of a particle $P$ , moving in a straight line, is $v\text{ m s}^{-1}$ at time $t$ seconds, where

$v = kt \quad \text{for } 0 \leqslant t \leqslant 4,$

$v = \tfrac16 k(t - 8)(t - 10) \quad \text{for } 4 \leqslant t \leqslant 10,$

and $k$ is a positive constant.
(a) Describe the graph of $v$ against $t$ for $0 \leqslant t \leqslant 10$ . Your description should include the shape of each section of the graph, the value of $v$ , in terms of $k$ , when $t = 4$ , and the values of $t$ at which the graph meets the $t$ -axis.

(3 marks)
(b) State the value of $t$ for which $\dfrac{dv}{dt} = 0$ .

(1 mark)
(c) In this question you must show detailed reasoning.

It is given that $P$ travels a total distance of $65$ m in the first 10 seconds of its motion.

Find the value of $k$ .

(6 marks)
Level 5 · 10 marksTry it