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

Assuming a constant power source influences the body's motion, the objective is to determine the relationship between displacement \( s \) and time \( t \).

Step 1: Power and Velocity Relationship
The constant power \( P \) supplied to the body is defined as:

\[ P = Fv, \]

where:
- \( F \) denotes the force acting on the body,
- \( v \) represents the body's velocity.

Applying Newton's second law, \( F = ma \) (where \( m \) is mass and \( a \) is acceleration), we get:

\[ P = mav. \]

Given that power is constant:

\[ P = mv \frac{dv}{dt}. \]

Step 2: Equation Integration
The equation is rearranged as:

\[ P \, dt = mv \, dv. \]

Integrating both sides yields:

\[ \int P \, dt = \int mv \, dv. \]

The result of the integration is:

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

Taking the square root gives the velocity:

\[ v = \sqrt{\frac{2Pt}{m}}. \]

Step 3: Displacement Calculation
Velocity is defined as the rate of change 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 leads to:

\[ s \propto t^{3/2}. \]

Consequently, displacement \( s \) is directly proportional to \( t^{3/2} \).