Question

A body starts from rest, under the action of an engine working under a constant power and moves along a straight line. The displacement S is given as a function of time (t) is given by:
(A)S=at+b${{t}^{2}}$ ; a, b are constants
(B)S=b${{t}^{2}}$ ; b is a constant
(C)S=a${{t}^{\dfrac{3}{2}}}$ ; a is a constant
(D)S=at; a is a constant

Answer 1

Hint: When an engine works under a constant power and moves along a straight line, the power must be considered constant in this case. Power is expressed as energy per unit time. Using this relation we proceed to find the relation between displacement and time.

Formula used: $Power=\dfrac{Energy}{Time}$

Complete step by step answer:
We have,
$Power=\dfrac{Energy}{Time}$
$\begin{align}
  & \Rightarrow P=\dfrac{\dfrac{1}{2}m{{v}^{2}}}{t} \\
 & \Rightarrow \text{constant}=\dfrac{{{v}^{2}}}{t} \\
 & \Rightarrow v=k{{t}^{\dfrac{1}{2}}} \\
 & \Rightarrow \dfrac{dx}{dt}=k{{t}^{\dfrac{1}{2}}} \\
 & \Rightarrow x=k\int{{{t}^{\dfrac{1}{2}}}}dt \\
 & \Rightarrow x=k{{t}^{\dfrac{3}{2}}} \\
 & \Rightarrow x\propto {{t}^{\dfrac{3}{2}}} \\
\end{align}$
Thus the dependence of time with displacement is given by the above relation which we obtain after the process of integration.
Here, k is a constant. So we can replace k by a then we will get our required solution.

So, the correct answer is “Option C”.

Additional Information: The quantity that deals with the amount of work done on or by a body is called power. In other words, power is the rate of doing work. Mathematically it can be defined in many ways. It may be defined as the work done per unit time, force multiplied by velocity. The standard unit of measuring work is Watt. All machines have a power rating in watt used to give us the value of total work done per unit time of the machine which is a useful tool that helps us determine the efficiency and utility of the machines.

Note: The problem can also be solved by taking power as force multiplied by velocity and then integrating both parts. Then the relationship between displacement and time can be established. The method of integration must be clear to solve this problem.