Question

A body is moving unidirectionally under the influence of a constant power source. Its displacement in time t is proportional to :

• A. t²
• B. $t^\frac{2}{3}$
• C. $t^\frac{3}{2}$
• D. t

Answer 1

The Correct Option is C

Given that the body moves under the influence of a constant power source, we aim to find the relation between the displacement $s$ and the time $t$.

Step 1: Understanding the Relationship Between Power and Velocity
Power $P$ delivered to the body is constant and is given by:

$$P = Fv,$$

where:
- $F$ is the force acting on the body,
- $v$ is the velocity of the body.

Using Newton’s second law $F = ma$, where $m$ is the mass and $a$ is the acceleration, we have:

$$P = mav.$$

Since power is constant, we can write:

$$P = mv \frac{dv}{dt}.$$

Step 2: Integrating the Equation
Rearranging:

$$P \, dt = mv \, dv.$$

Integrating both sides:

$$\int P \, dt = \int mv \, dv.$$

This yields:

$$Pt = \frac{mv^2}{2} \implies v^2 = \frac{2Pt}{m}.$$

Taking the square root:

$$v = \sqrt{\frac{2Pt}{m}}.$$

Step 3: Finding the Displacement
Velocity is the derivative of displacement with respect to time:

$$v = \frac{ds}{dt} = \sqrt{\frac{2Pt}{m}}.$$

Rearranging and integrating:

$$ds = \sqrt{\frac{2P}{m}} \, t^{1/2} \, dt.$$

Integrating both sides:

$$s \propto t^{3/2}.$$

Therefore, the displacement $s$ is proportional to $t^{3/2}$.