A-level · Kinematics

Motion graphs (displacement–time & velocity–time)Exam-style questions

What the gradient and the area mean on each kind of motion graph — gradient of displacement–time is velocity, gradient of velocity–time is acceleration, area under velocity–time is displacement — and how to model a journey as a graph and read unknowns off its area.

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

DifficultyAll Level 3 Level 4 Level 5

1
The points $A$ , $B$ and $C$ lie in that order on a straight horizontal running track. The point $B$ is on the finishing line of a race.

A runner passes $A$ with speed $4\ \text{m s}^{-1}$ . She accelerates uniformly from $A$ and crosses the finishing line at $B$ with speed $V\ \text{m s}^{-1}$ . After crossing the finishing line, she decelerates uniformly from $B$ and comes to rest at $C$ .

The runner is modelled as a particle.
(a) Sketch a speed-time graph for the motion of the runner from $A$ to $C$ .

(3 marks)

Given that $AB = 60$ m and $BC = 12$ m,
(b) find, in terms of $V$ ,

(i) the time taken by the runner to move from $A$ to $B$

(ii) the time taken by the runner to move from $B$ to $C$

(4 marks)

The runner takes $13$ s to move from $A$ to $C$ .
(c) Find the value of $V$ .

(2 marks)
(d) Find the deceleration of the runner between $B$ and $C$ .

(2 marks)
Level 4 · 11 marksTry it
2
A car is travelling along a straight horizontal road. It passes the point $O$ $O$ on the road with speed $5\ \text{m s}^{-1}$ $5 m s^{- 1}$ Figure 1 (not accurately drawn)

Figure 1 shows the acceleration-time graph for the motion of the car in the interval $0 \leqslant t \leqslant 35$ $0 ⩽ t ⩽ 35$ , where $t$ $t$ seconds is the time after the car passes $O$ $O$ . The acceleration of the car is
- $0.8\ \text{m s}^{-2}$ for $0 < t < 10$
- $-0.2\ \text{m s}^{-2}$ for $10 < t < 25$
- $-1\ \text{m s}^{-2}$ for $25 < t < 35$
The car is modelled as a particle.
(a) Show that the speed of the car when $t = 25$ $t = 25$ is $10\ \text{m s}^{-1}$ $10 m s^{- 1}$

(3 marks)
(b) Sketch a speed-time graph for the motion of the car in the interval $0 \leqslant t \leqslant 35$ $0 ⩽ t ⩽ 35$

(3 marks)
(c) Find the total distance travelled by the car in the interval $0 \leqslant t \leqslant 35$ $0 ⩽ t ⩽ 35$

(4 marks)
Level 4 · 10 marksTry it
3
Figure 1 — not accurately drawn

A racehorse runs a $1200$ $1200$ m race on a straight course, taking $72$ $72$ s from the start to the finishing line. It leaves the starting stalls from rest at time $t = 0$ $t = 0$

Figure 1 shows a speed-time graph that models the motion of the horse during the race.

The model assumes that the horse
- gains speed at a constant rate from $t = 0$ to $t = 6$ , when its speed is $18\ \text{m s}^{-1}$
- keeps this speed of $18\ \text{m s}^{-1}$ until $t = 51$
- loses speed at a constant rate from $t = 51$ until the end of the race, when its speed is $U\ \text{m s}^{-1}$
Using the model,
(a) find the acceleration of the horse while it is gaining speed, stating the units of your answer,

(2 marks)
(b) find the distance run by the horse in the first $51$ $51$ s of the race,

(3 marks)
(c) find the value of $U$ $U$ .

(3 marks)
Level 4 · 8 marksTry it
4
A coach and a car are travelling in the same direction along a straight horizontal motorway, in neighbouring lanes. The diagram shows the velocity–time graphs for their motion.The coach travels at a constant speed of $25\text{ m s}^{-1}$ . At time $t = 0$ seconds the coach passes the car, which is then moving at $15\text{ m s}^{-1}$ . The car accelerates uniformly until $t = 24$ , when its speed reaches $33\text{ m s}^{-1}$ , and it then continues at this constant speed.
(a) Calculate the acceleration of the car.

(2 marks)
(b) Determine the value of $t$ at which the car overtakes the coach.

(4 marks)
Level 4 · 6 marksTry it