15.2 The language of kinematics — Practise

Work through these on your own. Write down your positive direction before you start, and give every velocity and acceleration its sign. Check whether the question asks for a distance or a displacement.

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

A cyclist rides $4$ km east, then turns and rides $1.5$ km west. Taking east as positive, find the distance travelled and the displacement.
A runner jogs $2.4$ km along a straight path, then turns and jogs $0.8$ km back. Find the distance travelled and the displacement.
A car travels $420$ m along a straight road in $35$ s. Find its average speed.
A walker moves $300$ m due north in $20$ s. Find the average velocity.
A train moves $500$ m east in $25$ s, then $100$ m west in $20$ s. Find its average speed and its average velocity.
A car speeds up from $4$ m/s to $22$ m/s in $6$ s. Find its acceleration.
A train slows from $45$ m/s to $9$ m/s in $4$ s while travelling east. Taking east as positive, find its acceleration.
A ball is thrown vertically upwards at $14.7$ m/s and returns to the thrower's hand $3$ s later, moving downwards at $14.7$ m/s. Taking upwards as positive, find its acceleration.
A swimmer swims $25$ m to the far end of a straight pool, back to the start, then $25$ m to the far end again. Find the distance travelled and the displacement.
A car travels at an average speed of $20$ m/s for $15$ s. Find the distance travelled.
A particle on a straight line is at position $-3$ m when $t = 0$ and at position $7$ m when $t = 10$ s. Find its displacement and its average velocity.
A ball is dropped from rest. After $3$ s it is moving downwards at $29.4$ m/s. Taking downwards as positive, find its acceleration.

Answers and concise solutions

1. The distance and the displacement

Answer: Distance travelled $= 4 + 1.5 = 5.5$ km. Displacement $= (+4) + (-1.5) = +2.5$ km, which is $2.5$ km east.

2. The distance and the displacement

Answer: Distance travelled $= 2.4 + 0.8 = 3.2$ km. Displacement $= 2.4 - 0.8 = 1.6$ km forward along the path.

3. The average speed

Answer: Average speed $= \dfrac{\text{distance}}{\text{time}} = \dfrac{420}{35} = 12$ m/s.

4. The average velocity

Answer: Average velocity $= \dfrac{\text{displacement}}{\text{time}} = \dfrac{300}{20} = 15$ m/s north. The direction is part of the answer, because velocity is a vector.

5. The average speed and the average velocity

Answer: Total distance $= 600$ m over $45$ s, so the average speed is $\dfrac{600}{45} = \dfrac{40}{3} \approx 13.3$ m/s. The displacement is $400$ m east, so the average velocity is $\dfrac{400}{45} = \dfrac{80}{9} \approx 8.9$ m/s east.

6. The acceleration

Answer: Acceleration $= \dfrac{v - u}{t} = \dfrac{22 - 4}{6} = 3$ m/s $^2$ . The acceleration is positive, so the car speeds up in the positive direction.

7. The acceleration

Answer: Acceleration $= \dfrac{9 - 45}{4} = -9$ m/s $^2$ . The negative sign means the acceleration points west, opposite to the motion, so the train slows down.

8. The acceleration

Answer: Acceleration $= \dfrac{-14.7 - 14.7}{3} = \dfrac{-29.4}{3} = -9.8$ m/s $^2$ . The acceleration is $-9.8$ m/s $^2$ throughout, because gravity always acts downwards.

9. The distance and the displacement

Answer: Distance travelled $= 25 + 25 + 25 = 75$ m. The swimmer finishes $25$ m from the start, so the displacement is $25$ m.

10. The distance

Answer: Distance $=$ average speed $\times$ time $= 20 \times 15 = 300$ m.

11. The displacement and the average velocity

Answer: Displacement $= s_1 - s_0 = 7 - (-3) = 10$ m. Average velocity $= \dfrac{10}{10} = 1$ m/s in the positive direction.

12. The acceleration

Answer: Acceleration $= \dfrac{29.4 - 0}{3} = 9.8$ m/s $^2$ downwards. With downwards positive the acceleration is $+9.8$ m/s $^2$ , which is the acceleration of free fall.