A body is moving unidirectionally under the influence of a constant power source. Its displacement in time t is proportional to :
- t2
- \(t^\frac{2}{3}\)
- \(t^\frac{3}{2}\)
- t
The Correct Option is C
Solution and Explanation
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 \( t^{3/2} \).
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