Question

Among the four graphs, there is only one graph for which average velocity over the time interval \[\left( {0,T} \right)\] can vanish for a suitably chosen \[T\] . Which one is it?
A.

B.

C.

D.

Answer 1

Hint:First of all, we will look for a graph, which contains a positive and a negative as well, for a given time interval. We know that the summation of the positive and the negative displacement is zero.

Complete answer:
In the given question, we are given four graphs, which represent displacement versus time.
The vertical axis represents displacement and the horizontal axis represents time taken by the body. In the question, there is only one graph for which average velocity over the time interval \[\left( {0,T} \right)\] can vanish for a suitably chosen \[T\] .

To begin with, we will discuss a bit about the term called “average velocity”. Average velocity is defined as the ratio of the displacement of a body to the time taken by it to cover the given displacement.
Let us proceed to solve the problem.
We need to describe the graph, which has the same displacement for two timings, according to this problem. The related speeds should be in opposite directions, if there are two timings with the same displacement. The first slope, as seen in graph (B), decreases, which suggests that the particle moves in one direction and decreases its velocity, becomes zero at the highest point of the curve and then increases in the backward direction. The particle would then return to its original position.
So, there are two distinct time points with one value of displacement and we realise that the \[x\] , \[x - t\] graph slope gives us the average velocity. Slope is therefore positive for one time, therefore average velocity is also positive for \[A\] , and slope is negative for another time, then average velocity is also negative. Because in the \[O\] to \[T\] interval there are opposite velocities, average velocity can also only vanish in choice (B).

This can be seen in the corresponding figure given.

As seen in the diagram, for two separate time points, \[OA = BT\] (same displacement).

The correct option is B.

Note:While answering this question, many students tend to make mistakes by not taking the direction into account. It is important to remember that if you move from one position to another and return back to the original position again, then the displacement is zero. This is because the initial and the final position is the same. However, the distance covered is the summation of the distance covered in the two journeys.