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Distance and displacement tell us how far. But how do we capture how fast an object moves — and how do we add direction to that?
- The average speed is the total distance travelled divided by the time interval taken: $$\text{average speed} = \dfrac{\text{total distance travelled}}{\text{time interval}}$$
- Since distance has no direction, average speed has only a numerical value (no direction).
- Uniform motion: equal distances in equal intervals of time (constant speed). Non-uniform motion: unequal distances in equal intervals (speed increasing/decreasing).
- The idea that speed = distance ÷ time is ancient in India, seen in the Aryabhatiya (5th century CE). The problem below is from the Ganitakaumudi (14th century CE).
Two postmen start 210 yojanas apart and walk towards each other. One covers 9 yojanas /day, the other 5 yojanas /day. In how many days do they meet?
Distance closed per day \( = 9 + 5 = 14 \) yojanas .
Days to cover 210 together \( = \dfrac{210}{14} = 15 \) days.
So they meet after 15 days (the first covers 135 yojanas , the second 75 yojanas ).
- Average velocity = change in position (displacement) ÷ time interval: $$v_{av} = \dfrac{\text{displacement}}{\text{time interval}} = \dfrac{s}{t}$$
- It needs magnitude and direction ; the direction is the same as that of the displacement (shown by + or −).
- SI unit of both average speed and average velocity is metre per second (m s⁻¹ or m/s); also km h⁻¹.
- Average velocity is the average rate of change of position with respect to time.
- Average speed — the total distance travelled divided by the time interval during which the distance is covered.
- Average velocity — the change in position (displacement) divided by the time interval in which the change occurs.
- Uniform motion — motion in which equal distances are covered in equal intervals of time (constant speed).
- Non-uniform motion — motion in which unequal distances are covered in equal intervals of time (changing speed).
- Rate of change — the ratio of the change in one quantity to the corresponding change in time; average velocity is the rate of change of position with time.
- The velocity at a particular instant is called the instantaneous velocity .
- As the time interval around an instant is made smaller and smaller, the average velocity approaches a fixed value — that value is the instantaneous velocity. Throughout this chapter, ‘velocity’ means the velocity at a particular instant.
- For motion in a straight line, the average speed and the magnitude of average velocity in a time interval are equal if the object moves in one direction .
Sarang swims from one end of a 25 m pool to the other and back in 50 s. Find his average speed and average velocity.
Total distance \( = 50\text{ m} \), displacement \( = 0\text{ m} \) (he returns to start).
\( \text{average speed} = \dfrac{50\text{ m}}{50\text{ s}} = 1\ \text{m s}^{-1} \)
\( \text{average velocity} = \dfrac{0\text{ m}}{50\text{ s}} = 0\ \text{m s}^{-1} \)
So average speed is about \(1\ \text{m s}^{-1}\) while average velocity is \(0\ \text{m s}^{-1}\).
- 4. Road trip: 200 km north in 3 h, then 200 km south in 2 h. Total distance 400 km in 5 h → average speed \(= \dfrac{400}{5} = 80\ \text{km h}^{-1}\). Net displacement \(= 0\) → average velocity \(= 0\).
- 5. Conditions: (i) magnitude of average velocity equals average speed when the object moves in one direction without turning back. (ii) average velocity is zero while average speed is not, when the object returns to its starting point (displacement 0, path length non-zero).
