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 $P$ moves on the $x$ -axis. When $t = 0$ , $P$ is at the origin $O$ and is at rest. At time $t$ seconds, $P$ has acceleration $(k - 4t)$ m s $^{-2}$ in the direction of increasing $x$ , where $k$ is a positive constant.

Three seconds after it starts to move, $P$ has speed $18$ m s $^{-1}$ .
(a) Show that $k = 12$

(3 marks)

The displacement of $P$ from $O$ at time $t$ seconds is $x$ metres.
(b) Show that $x = \frac{2}{3}t^2(9 - t)$

(2 marks)
(c) Find the total distance travelled by $P$ from the time it starts to move until it is next at $O$ .

(4 marks)
Level 4 · 9 marksTry it
2
A particle $P$ moves along a straight line. At time $t$ seconds, where $t \geqslant 0$ , the velocity of $P$ is $v$ m s⁻¹, where

$v = 3t^2 - 12t + 9$

At $t = 0$ , $P$ is at the point $O$ on the line.
(a) Find the values of $t$ at which $P$ is instantaneously at rest. (2 marks)
(b) Find the total distance travelled by $P$ in the first $4$ seconds. (6 marks)
(c) Find the magnitude of the acceleration of $P$ when $t = 4$ . (2 marks)
Level 4 · 10 marksTry it
3
A particle $P$ moves along a straight line. At time $t$ seconds, where $t \geqslant 0$ , the acceleration of $P$ is $a$ m s⁻², where

$a = 6t - 4$

When $t = 0$ , $P$ is at the origin $O$ and is moving with velocity $5$ m s⁻¹.
(a) Find an expression for the velocity of $P$ at time $t$ . (2 marks)
(b) Find the displacement of $P$ from $O$ when $t = 3$ . (3 marks)
(c) Show that $P$ never comes to rest. (2 marks)
Level 4 · 7 marksTry it
4
A cyclist rides along a straight, horizontal cycle path.

$O$ is a fixed point on the path.

At time $t$ seconds the displacement of the cyclist from $O$ is $x$ metres, where

$x = p + 8t + q\text{e}^{-0.25t}$

and $p$ and $q$ are constants.

The cyclist's acceleration is $0.4\ \text{m s}^{-2}$ at the instant when $t = 4$
(a) Show that $q \approx 17.4$

(5 marks)
(b) When $t = 0$ the cyclist is $25$ metres from $O$ , and $x > 0$

Find the value of $p$

Give your answer to two significant figures.

(2 marks)
Level 4 · 7 marksTry it
5
A remote-controlled model car $C$ , of mass $1.5$ kg, is driven across a flat horizontal playground. The car is modelled as a particle. At time $t$ seconds, where $t \geq 0$ , its velocity is $\mathbf{v}\ \text{m s}^{-1}$ , where

$\mathbf{v} = (6 - t)\mathbf{i} + (t^2 - 2t + 4)\mathbf{j},$

and the resultant horizontal force acting on $C$ is $\mathbf{F}$ N.
(a) Show that $C$ is never stationary.

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

(1 mark)
(c) Determine the magnitude of $\mathbf{F}$ at the instant when $C$ is moving in the direction of the vector $2\mathbf{i} + 3\mathbf{j}$ . Give your answer correct to 3 significant figures.

(5 marks)
Level 4 · 8 marksTry it

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