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💬 Slope and area
A velocity-time graph hides two treasures: its slope gives acceleration, and the area under it gives displacement. How?
What does the shape of a velocity-time graph tell us?
- Line parallel to time axis (Fig. 4.17a): velocity constant → acceleration is zero .
- Rising straight line (Fig. 4.17b): velocity increasing by equal amounts → constant acceleration in the direction of velocity.
- Falling straight line (Fig. 4.17c): velocity decreasing by equal amounts → constant acceleration opposite to velocity.
Important Definitions
- Velocity-time graph — a graph that represents the change in the velocity of an object with time; its slope gives acceleration and the area under it gives displacement.
| Time | Velocity of car |
|---|---|
| 0 s | 0 m s⁻¹ |
| 5 s | 2.5 m s⁻¹ |
| 10 s | 5.0 m s⁻¹ |
| 15 s | 7.5 m s⁻¹ |
| 20 s | 10.0 m s⁻¹ |
| 25 s | 12.5 m s⁻¹ |
| 30 s | 15.0 m s⁻¹ |
| Time | Velocity of car |
|---|---|
| 0 s | 15.0 m s⁻¹ |
| 5 s | 12.5 m s⁻¹ |
| 10 s | 10.0 m s⁻¹ |
| 15 s | 7.5 m s⁻¹ |
| 20 s | 5.0 m s⁻¹ |
| 25 s | 2.5 m s⁻¹ |
| 30 s | 0 m s⁻¹ |
How do we get acceleration and displacement from a velocity-time graph?
- Slope = acceleration. For part AB: \( a = \dfrac{v - u}{t_2 - t_1} = \dfrac{10 - 5}{20 - 10} = 0.5\ \text{m s}^{-2} \). A falling line gives a negative slope (e.g. −0.5 m s⁻²).
- Area under the line = displacement. Constant velocity: displacement \(= 20\ \text{m s}^{-1} \times 6\ \text{s} = 120\ \text{m}\).
- For a sloping line, split the area into a rectangle + triangle: $$s = (\text{rectangle}) + \tfrac{1}{2}(\text{triangle}) = 50 + 25 = 75\ \text{m}$$ (displacement between 10 s and 20 s in Fig. 4.18b).
What a velocity-time graph gives you
Velocity-time graph
velocity plotted against time
↓
Slope of the line
gives the acceleration
↓
Area between the line and the time axis = displacement
NCERT Question 12 — The velocity-time graph from 0
For a cyclist’s v-t graph (0–120 s, Fig. 4.30): shade the displacement in the constant and decreasing parts, then find displacement and average acceleration.
View the answer →NCERT Question 13 — A girl is preparing for
From a marathon runner’s velocity-time graph (Fig. 4.31), estimate the distance she ran.
View the answer →NCERT Question 14 — On entering a state highway
A car goes 6 m s⁻¹ for 2 min, then accelerates at 1 m s⁻² for 6 s. Find the displacement using a velocity-time graph.
View the answer →NCERT Question 15 — Two cars A and B
Car A reaches 5 m s⁻¹ in 5 s, Car B reaches 3 m s⁻¹ in 10 s, both from rest. Plot v-t graphs and find the displacements.
View the answer →