Question

Given the velocity-time graph, how can it be used to find the distance of a body in a given time?
(A) The total area under the velocity-time graph
(B) The net area under the velocity-time graph
(C) Slope of velocity-time graph
(D) Negative slope of velocity-time graph

Answer 1

Here, a graphical representation of velocity with time is given and you are asked to find a way in which we can obtain the distance of the body at a given time t. Firstly, the net area under a graph means that the area under the graph is added considering the sign. Total area gives you the sum of magnitude of positive area and negative area. In order to answer this question, you need to consider the above-mentioned information and the definition of velocity of a body.

Complete answer:
Suppose a graph is given to you which represents the variation of velocity with respect to time. The velocity might be positive or negative. Velocity of an object or a body is defined as the rate of change of position of the body with respect to time. Mathematically, we have,
$v = \dfrac{{dx}}{{dt}}$, where $v$ is the velocity (derivative of displacement with respect to time). Now let us rearrange the above equation, we get,$dx = vdt$. In order to get the displacement, you will need to integrate the equation. Let the initial conditions be as follows, at $t = 0,x = 0$
\int\limits_0^x {dx} = \int {vdt} \\\ \therefore x = \int {vdt} \\\
As you can see that the displacement of the body is the net area under the velocity-time graph. $v$ being the y-coordinate and $dt$ being the infinitesimally thin strip together making a rectangle and after getting integrated will give you the net area under the graph.

If you need the distance of the body, then you must consider the magnitude of the velocity and not the sign. Considering signs will lead to wrong answers.Hence, the total area under the velocity-time graph can be used to find the distance of a body in a given time.

Hence, option A is correct.

Note: You need to remember that velocity is the rate of change of displacement of the body with respect to time or derivative of displacement with respect to time. After that, you can simply integrate velocity with respect to time to find the displacement of the body. The slope of velocity-graph will give you the acceleration of the body. Keep in mind the difference between net and total area under the graph.