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 \( t^{3/2} \).