3.1 Speed, velocity and motion graphs — Practise

Work through these on your own. Keep your working neat: the method matters as much as the answer.

Both the questions and the answers below come from a set that is checked by computer every time the book is built.

A cyclist travels 180 m in 30 s. Calculate her average speed.
A car travels at a steady 15 m/s for 20 s. Calculate the distance it travels.
A train travels 900 m at a steady 30 m/s. Calculate the time taken.
A walker covers 1200 m in 800 s. Calculate his average speed.
A car speeds up from 8.0 m/s to 20 m/s in 4.0 s. Calculate its acceleration.
A bus slows from 15 m/s to rest in 5.0 s. Calculate the size of its deceleration.
A distance-time graph is a straight line through $(0\,\text{s}, 0\,\text{m})$ , $(10\,\text{s}, 60\,\text{m})$ and $(20\,\text{s}, 120\,\text{m})$ . Calculate the speed from the gradient of the line.
A velocity-time graph is a straight line through $(0\,\text{s}, 0\,\text{m/s})$ , $(4.0\,\text{s}, 10\,\text{m/s})$ and $(8.0\,\text{s}, 20\,\text{m/s})$ . Calculate the acceleration from the gradient of the line.
A car travels at 24 m/s. The driver brakes with a steady deceleration of 6.0 m/s² until the car stops. Calculate the braking distance.
A trolley starts from rest and accelerates at 2.0 m/s² over a distance of 25 m. Calculate its final velocity.
H A distance-time graph for an accelerating object is curved. The tangent at 4.0 s passes through $(2.0\,\text{s}, 10\,\text{m})$ and $(6.0\,\text{s}, 30\,\text{m})$ . Calculate the speed at 4.0 s.
H A velocity-time graph rises in a straight line from 0 m/s at 0 s to 20 m/s at 8.0 s. Calculate the distance travelled from the area under the graph.
H A car travels around a roundabout at a steady 8.0 m/s. Explain why the velocity of the car is changing even though its speed is constant.
Describe the difference between distance and displacement, and between speed and velocity.

Answers and concise solutions

1. Average speed of the cyclist

Answer: $v = \dfrac{s}{t} = \dfrac{180}{30} = 6.0\,\text{m/s}$ .

2. Distance travelled

Answer: $s = vt = 15 \times 20 = 300\,\text{m} = 3.0 \times 10^2\,\text{m}$ to 2 significant figures.

3. Time taken by the train

Answer: $t = \dfrac{s}{v} = \dfrac{900}{30} = 30\,\text{s} = 3.0 \times 10^1\,\text{s}$ to 2 significant figures.

4. Average speed of the walker

Answer: $v = \dfrac{s}{t} = \dfrac{1200}{800} = 1.5\,\text{m/s}$ . This is the typical speed of a person walking.

5. Acceleration of the car

Answer: $a = \dfrac{\Delta v}{t} = \dfrac{20 - 8.0}{4.0} = 3.0\,\text{m/s}^2$ .

6. Deceleration of the bus

Answer: The acceleration is $\dfrac{0 - 15}{5.0} = -3.0\,\text{m/s}^2$ , so the deceleration is $3.0\,\text{m/s}^2$ .

7. Speed from the gradient

Answer: $\text{speed} = \dfrac{120 - 0}{20 - 0} = 6.0\,\text{m/s}$ . The gradient of a distance-time graph is the speed.

8. Acceleration from the gradient

Answer: $a = \dfrac{20 - 0}{8.0 - 0} = 2.5\,\text{m/s}^2$ . The gradient of a velocity-time graph is the acceleration.

9. Braking distance

Answer: $v^2 - u^2 = 2as$ , and $v = 0$ , so $s = \dfrac{u^2}{2a} = \dfrac{24^2}{2 \times 6.0} = \dfrac{576}{12} = 48\,\text{m}$ .

10. Final velocity of the trolley

Answer: $v^2 - u^2 = 2as$ with $u = 0$ , so $v^2 = 2 \times 2.0 \times 25 = 100$ and $v = 10\,\text{m/s} = 1.0 \times 10^1\,\text{m/s}$ .

11. Speed from the tangent

Answer: The gradient of the tangent is the speed at that moment: $\dfrac{30 - 10}{6.0 - 2.0} = 5.0\,\text{m/s}$ .

12. Distance from the area under the graph

Answer: The area is a triangle: $\tfrac{1}{2} \times 8.0 \times 20 = 80\,\text{m}$ . The area under a velocity-time graph is the distance travelled.

13. Why the velocity is changing

The speed is constant, because the car covers the same distance each second.
The direction of travel changes all the way around the roundabout, and velocity is speed in a given direction, so the velocity changes.

14. Distance and displacement, speed and velocity

Distance is how far an object moves and has no direction, so it is a scalar. Displacement is the distance in a straight line from the start to the finish, with its direction, so it is a vector.
Speed is how fast an object moves and has no direction, so it is a scalar. Velocity is speed in a given direction, so it is a vector.