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A velocity-time graph hides two treasures: its slope gives acceleration, and the area under it gives displacement. How?
- 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.
- 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 |
- Slope = acceleration. For part AB: a = v - u t 2 - t 1 = 10 - 5 20 - 10 = 0.5 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 m s -1 × 6 s = 120 m .
- For a sloping line, split the area into a rectangle + triangle: s = ( Rectangle ) + 1 2 ( Triangle ) = 50 + 25 = 75 m (displacement between 10 s and 20 s in Fig. 4.18b).
NCERT Question 12 — The velocity-time graph from 0
The velocity-time graph from 0 s to 120 s for a cyclist is shown in Fig. 4.30. Shade the areas (in different colours) representing the displacement of the cyclist
(i) while cyclist is moving with constant velocity.
(ii) when the velocity of cyclist is decreasing.
Also, calculate the displacement and average acceleration in the 120 s time interval.
NCERT Question 13 — A girl is preparing for
A girl is preparing for her first marathon by running on a straight road. She uses a smartwatch to calculate her running speed at different intervals. The graph (Fig. 4.31) depicts her velocity versus time. Estimate the distance she ran based on the graph.
NCERT Question 14 — On entering a state highway
On entering a state highway, a car continues to move with a constant velocity of 6 m s⁻¹ for 2 minutes and then accelerates with a constant acceleration 1 m s⁻² for 6 seconds. Find the displacement of the car on the state highway in the 2 min 6 s time interval by drawing a velocity-time graph for its motion.
NCERT Question 15 — Two cars A and B
Two cars A and B start moving with a constant acceleration from rest, in a straight line. Car A attains a velocity of 5 m s⁻¹ in 5 s. Car B attains a velocity of 3 m s⁻¹ in 10 s. Plot the velocity-time graphs for both the cars in the same graph. Using the graph, calculate the displacement in the two time intervals mentioned.
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What does the slope of a position-time graph represent?
Show Answer
The velocity of the object. A steeper slope means a larger velocity; a horizontal line means the object is at rest. -
What does the slope of a velocity-time graph represent?
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The acceleration of the object. -
What does the area under a velocity-time graph represent?
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The displacement of the object in that time interval. -
A position-time graph is a straight line sloping downwards. What is the motion?
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The object moves with a constant velocity in the negative direction (position decreases uniformly with time).
