Question

A body is moved along a straight line by a machine delivering constant power. The distance travelled by the body in time t is proportional to?
(A) ${t^{1/2}}$
(B) $t$
(C) ${t^{3/2}}$
(D) ${t^2}$

Answer 1

Hint
It may be assumed that the distance travelled by the body (s) is proportional to nth power of time t. Then it is to be differentiated two times to derive the expression for force. The force can then be used to derive the formula for Power. In the question, it is mentioned that the power is constant, so the expression for the power can be equated to a constant value. And the value of n can be calculated.

Complete step by step answer
Assuming that the distance travelled by body is proportional to ${t^n}$,
We can write,
$\Rightarrow s = k{t^n}$ where k is a constant value.
On differentiating with respect to t,
$\Rightarrow v = \dfrac{{ds}}{{dt}} = nk{t^{n - 1}}$ where v is the velocity of the body.
On differentiating again,
$\Rightarrow a = \dfrac{{dv}}{{dt}} = n(n - 1)k{t^{n - 2}}$ where a is the acceleration of the body.
We know that $F = ma$, (Definition of force)
Substituting the value of $a$ in the above equation,
$\Rightarrow F = mn(n - 1)k{t^{n - 2}}$
Now power(P) is defined as product of force and velocity, keeping values of F and v,
$\Rightarrow P = F.v = mn(n - 1)k{t^{n - 2}}.nk{t^{n - 1}}$
$\Rightarrow P = m{n^2}{k^2}(n - 1){t^{2n - 3}}$
We know from question that $P = {\text{constant}}$,
$\therefore P$ is independent of time.
Which means the power of the term containing t should be 0.
$\Rightarrow 2n - 3 = 0$
From here the value of n can be calculated,
$\Rightarrow n = \dfrac{3}{2}$
$\therefore s \propto {t^{3/2}}$
Hence the correct option is option (C).

Note
This can also be solved using an alternate method where we start from writing the power as product of F and v, and then integrating the value of v two times to get s,
The final equation will look like:
$\Rightarrow s = \sqrt {\dfrac{{2P}}{m}} \int_0^t {\sqrt t dt}$
Now if t is integrated, we get,
$\Rightarrow s = \dfrac{2}{3}\sqrt {\dfrac{{2P}}{m}} .{t^{3/2}}$
If this equation is differentiated we get,
$\Rightarrow {\text{constant = }}\dfrac{2}{3}\sqrt {\dfrac{{2Pt}}{m}}$
Which implies, $s \propto {t^{3/2}}$.