# 2.1 Motion (I)

F.4 Physics · Topic II Force and Motion · Lesson 1

Distance and displacement, speed and velocity, acceleration — and which of them need a direction.

## Figure

Sign of the acceleration for motion in a straight line, with forward taken as positive.

## Key points

- **Scalars ／ 純量** have **magnitude ／ 量值** only; **vectors ／ 向量** have magnitude **and** direction.
- **Distance** = total length of the **actual path**. **Displacement ／ 位移** = length of the **shortest path** from start to end, **with direction**. Distance is greater, except in a straight line one way.
- **Speed** = distance per unit time; **velocity ／ 速度** = displacement per unit time, in the same direction as the displacement.
- **Average** = total ÷ total time. **Instantaneous** = the value at one instant (speedometer, motion sensor).
- **Acceleration ／ 加速度** = change in velocity per unit time. Speeding up, slowing down **and turning a corner at constant speed** are all acceleration.

## Must know

| Quantity | Type | Average formula | SI unit |
|---|---|---|---|
| Distance $d$ | scalar | — | m |
| Displacement $s$ | vector | — | m |
| Speed $v$ | scalar | $\bar{v} = \dfrac{d}{t}$ | m s⁻¹ |
| Velocity $v$ | vector | $\bar{v} = \dfrac{s}{t}$ | m s⁻¹ |
| Acceleration $a$ | vector | $\bar{a} = \dfrac{v - u}{t}$ | m s⁻² |

Conversion: $1\ \text{km h}^{-1} = \dfrac{1}{3.6}\ \text{m s}^{-1}$, so $100\ \text{km h}^{-1} = 27.8\ \text{m s}^{-1}$.

## Common mistakes

- ✗ "Displacement = 36.1 m." ✓ A vector answer needs a direction: "36.1 m, N33.7°E", or a + / − sign in a straight line.
- ✗ Averaging the speeds: $(2 + 3) \div 2 = 2.5$ m s⁻¹. ✓ Average speed = **total distance ÷ total time**: out at 2 and back at 3 m s⁻¹ gives 2.4 m s⁻¹, as the legs take different times.
- ✗ "Average speed = magnitude of the average velocity." ✓ Equal only in a straight line without turning back. Round a closed loop the average velocity is **zero**.
- ✗ "Constant speed means no acceleration." ✓ Zero acceleration needs constant **velocity**. Rounding a curve at constant speed is acceleration: the direction changes.
- ✗ "$a$ is negative because the body is slowing down." ✓ The sign of $a$ is its **direction**. A body slows down whenever $a$ and $v$ have **opposite** signs.

## Quick check

1. A man walks 40 m due east, then 30 m due north. Find the total distance travelled and the magnitude of his displacement.
2. A car travels 300 m in 20 s, then a further 100 m in 30 s along the same straight road without turning back. Find its average speed.
3. A car moving forward slows from 20 m s⁻¹ to 8 m s⁻¹ in 4 s. Find its acceleration and say what the sign means.

## Answers

1. $d = 40 + 30 = 70$ m; $|s| = \sqrt{40^{2} + 30^{2}} = 50$ m (N53.1°E).
2. $\bar{v} = \dfrac{300 + 100}{20 + 30} = \dfrac{400}{50} = 8$ m s⁻¹.
3. $\bar{a} = \dfrac{8 - 20}{4} = -3$ m s⁻². The minus sign means $a$ points backward, opposite to the forward velocity, so the car slows down.

## Did you know?

Yang Liwei, China's first astronaut, circled the Earth 14 times in about 21 hours on Shenzhou-5 in 2003. He travelled nearly 600 000 km, yet his displacement at landing was under 1 000 km.