Question

A particle is moving unidirectionally on a horizontal plane under the action of a constant power-supplying energy source. Which of the following graphs accurately represents the motion of the particle in the displacement (s) – time (t) graph? (Graphs are drawn schematically and are not to scale)
A.
B.
C.
D.

Answer 1

Hint:Before solving this problem let’s understand the term displacement. When a force is applied, the object changes its position which is known as displacement. Since it is a vector quantity it has both direction and magnitude. The power supplied to the particle is the product of the force applied and the velocity at which it travels.

Formula Used:
The power supplied to the particle is,
\[P = FV\]
Where, F is force applied and V is velocity.

Complete step by step solution:
Consider a particle that is moving in one direction on a horizontal plane under the action of a constant power-supplying energy source. We need to find which of the following graphs accurately represents the motion of the particle in the displacement- time graph. We know that the power supplied to the particle is,
\[P = FV\]
We know that, \[F = m\dfrac{{dV}}{{dt}}\]
\[P = m\dfrac{{dV}}{{dt}} \times V\]
\[\Rightarrow VdV = \dfrac{P}{m}dt\]

On integrating we get the expression for V
\[\dfrac{{{V^2}}}{2} = \dfrac{P}{m}t\]
This clearly says that,
\[{V^2} \propto t\] or \[V \propto {t^{\dfrac{1}{2}}}\]
Here, velocity is,
\[V = \dfrac{\text{displacement}}{\text{time}}\]
\[ \Rightarrow V = \dfrac{S}{t}\]
Then,
\[\dfrac{S}{t} \propto {t^{\dfrac{1}{2}}}\]
\[\therefore S \propto {t^{\dfrac{3}{2}}}\]
Therefore, we represent this equation in the displacement time graph as shown in option B.

Hence, Option B is the correct answer.

Note:The displacement-time graphs show how the displacement of a moving object changes with time. If a displacement-time graph is a sloping line then it shows that the object is moving. In this graph, the slope or gradient of the line is equal to the velocity of the object.