Question

Area under velocity-time graph gives

• A. displacement
• B. work
• C. acceleration
• D. moment

Hint:

In kinematics, the area under the velocity-time graph represents displacement, while the area under a force-distance graph represents work.

Answer 1

The Correct Option is A

Displacement is defined mathematically as the integral of velocity with respect to time: \( x = \int v \, dt \). This integral is, by definition, exactly the area enclosed between the velocity curve and the time axis. So evaluating each option by this definition:

1. Displacement: Matches directly, since \( \int v\,dt \) is precisely how displacement is defined and precisely what "area under the curve" means.
2. Work: Work is defined as \( \int F\,dx \), the integral of force with respect to displacement, not velocity with respect to time. There is no way to recover work from a v-t graph alone.
3. Acceleration: Acceleration is \( \dfrac{dv}{dt} \), a derivative (the slope of the graph), which is the opposite operation of finding an area (an integral).
4. Moment: Moment involves force acting at a perpendicular distance, which has no relationship to a velocity-time plot at all.

By the calculus definition of displacement as \( \int v\,dt \), the area under the velocity-time graph is displacement.

Therefore, the correct answer is displacement.

Answer 2

A concrete example makes this clear. Suppose a body moves at a constant velocity of 5 m/s for 4 seconds. On the velocity-time graph, this is a horizontal line at height 5, and the "area under the graph" is the rectangle \( 5 \times 4 = 20 \). Physically, the body has moved \( 5 \, \text{m/s} \times 4 \, \text{s} = 20 \, \text{m} \) — exactly its displacement. Let's test each option against this worked example.

1. Displacement: The body's actual displacement in this example is 20 m, exactly matching the rectangle's area. This is confirmed by direct example.
2. Work: Computing work would require knowing the force involved (and the body's mass), none of which is given in a velocity-time plot, so a numeric value for work cannot even be obtained from this graph.
3. Acceleration: In this example, the velocity is constant, so the acceleration is zero throughout — but the area (20) is clearly not zero, showing area and acceleration are unrelated quantities here.
4. Moment: There's no rotational axis or force distance involved in this straight-line motion example, so "moment" has no meaningful value to compare against the area at all.

This worked example confirms that the area under a velocity-time graph gives the displacement.

Therefore, the correct answer is displacement.