Question

The area under velocity-time graph gives:
A. acceleration
B. distance
C. displacement
D. velocity

Answer 1

Use the formula for velocity of an object for a small displacement. This formula gives the relation between the small displacement of the object and time interval required for this travel. Rearrange this formula for small displacement. Take integration of this equation with respect to time and determine what is given by the area under the velocity-time graph.

Formula used:
The velocity $v$ of an object is given by
$v = \dfrac{{ds}}{{dt}}$ …… (1)
Here, $ds$ is the small change in displacement of the object and $dt$ is the time interval during which the object covers the same displacement.

Complete step by step solution:
In a velocity-time graph, the velocity of the object is plotted usually on Y-axis and time is plotted usually on X-axis.Rearrange equation (1) for the small displacement of the object.
$ds = vdt$
Take integration on both sides of the above equation.
$\int {ds} = \int {vdt}$
The integration of velocity with respect to time of an object at a particular time on a velocity-time graph gives the area under the curve. Hence, the right hand side of the above equation gives the area under the velocity-time graph.

Also, the integration of velocity of the object with respect to time is equal to displacement of the object.The integration of the small displacement over the determined limits gives the total displacement $s$ of the object.
Substitute $s$ for $\int {ds}$ and ${\text{Area}}$ for $\int {vdt}$ in the above equation.
$s = {\text{Area}}$
$\therefore {\text{Area}} = s$
This proves that the area under the velocity-time graph gives displacement.

Hence, the correct option is C.

Note: The students should always remember that the integration of velocity of an object with respect to time always gives the area under the velocity-time graph of the same object just like the integration of acceleration of the an object with respect to time gives the area under acceleration-time graph of the object.