Speed

1. Definition

Speed is the distance travelled by an object per unit time, regardless of the direction of motion.

[\text{Speed} = \dfrac{\text{Distance Travelled}}{\text{Time Taken}}]

Speed is a scalar quantity — it has only magnitude, no direction.

2. SI Unit

[\text{SI Unit of Speed} = \text{metre per second (m·s}^{-1})]

3. Dimensional Formula

[\text{Speed}] = [M^{0} L^{1} T^{-1}]

4. Types of Speed

Uniform Speed:
An object is said to have uniform speed if it covers equal distances in equal intervals of time, no matter how small the intervals are.
Non-Uniform (Variable) Speed:
An object has variable speed if it covers unequal distances in equal intervals of time.
Average Speed:
When an object moves with variable speed, its average speed over a time interval is given by:
[v_{avg} = \dfrac{\text{Total Distance Travelled}}{\text{Total Time Taken}}]
Instantaneous Speed:
The speed of an object at a particular instant of time.
Mathematically, [v = \lim_{\Delta t \to 0} \dfrac{\Delta s}{\Delta t} = \dfrac{ds}{dt}]

5. Key Points

Speed is always non-negative.
It depends on the total path length, not on the direction of motion.
For rectilinear motion in one direction without reversal:

[\text{Speed} = \text{Magnitude of Velocity}]

6. Practice Questions

Solved Examples

Q1. A car covers 120 km in 2 hours. Find its speed.
Solution:
[\text{Speed} = \dfrac{\text{Distance}}{\text{Time}} = \dfrac{120 \text{km}}{2 \text{h}} = 60 \text{km/h}]

Q2. A boy runs 300 m in 40 s. Find his speed in m/s and km/h.
Solution:
[\text{Speed} = \dfrac{300\text{m}}{40\text{s}} = 7.5 \text{m/s}]
To convert to km/h:
[7.5 \times 3.6 = 27 ,\text{km/h}]

Q3. A cyclist moves with speeds of 20 km/h for the first hour and 30 km/h for the next hour. Find the average speed.
Solution:
Total distance = [20 × 1 + 30 × 1 = 50 \text{km}]
Total time = 2 h
[v_{avg} = \dfrac{50}{2} = 25 \text{km/h}]

Q4. A car covers first half of the distance at 40 km/h and second half at 60 km/h. Find the average speed.
Solution:
Let total distance = 2d → each half = d.

Time taken in first half: [t₁ = \dfrac{d}{40}]
Time taken in second half: [t₂ = \dfrac{d}{60}]

Total distance = [2d], Total time = [\dfrac{d}{40} + \dfrac{d}{60} = \dfrac{5d}{120}]

[v_{avg} = \dfrac{2d}{(5d/120)} = 48 \text{km/h}]

Conceptual Questions (With Answers)

1. Define speed. Is it scalar or vector?

Speed is the distance travelled per unit time. It is a scalar quantity.

2. What is the SI unit and dimensional formula of speed?

SI Unit: m·s⁻¹
Dimensional Formula: [M⁰ L¹ T⁻¹]

3. Differentiate between uniform and non-uniform speed.

Uniform speed: Equal distances in equal time intervals.
Non-uniform speed: Unequal distances in equal time intervals.

4. Define average speed. How is it calculated?

Average speed = (Total distance travelled) ÷ (Total time taken)

5. Define instantaneous speed.

Speed of an object at a particular instant of time, v = ds/dt.

6. Can speed be negative?

No. Speed is always non-negative.

7. Is average speed always equal to the arithmetic mean of initial and final speeds?

No. This is true only when the object travels equal time intervals at both speeds, not when it covers equal distances.

8. For uniform rectilinear motion, how are speed and velocity related?

Speed = magnitude of velocity.

9. A car moves in a circular track with constant speed. Is its velocity constant?

No. Although speed is constant, the direction of velocity changes continuously, so velocity is variable.

10. A body starts from rest, moves 10 m in 2 s. What is its average speed?

Average speed = 10 m ÷ 2 s = 5 m/s.

7. Common Misconceptions / FAQs

1. If speed is constant, does it mean velocity is constant?

Not necessarily. Velocity also depends on direction; if direction changes, velocity changes.

2. Can average speed ever be zero?

Only if the total distance travelled is zero — which means the object never moved.

3. Can instantaneous speed be greater than average speed?

Yes. For variable speed motion, instantaneous speed can be higher or lower than the average.

4. Does zero velocity imply zero speed?

Zero velocity means no displacement per unit time. If the object is stationary, both are zero.