Question

A body is moving along a straight line under the influence of a constant power source. If the relation between the displacement (s) of the body and time (t) is \(s \propto t^x\), then \(x =\)

• A. \(\frac{1}{2}\)
• B. \(2\)
• C. \(\frac{2}{3}\)
• D. \(\frac{3}{2}\)

Hint:

When power is constant, use the relationship \(P = Fv\) and express everything in terms of \(t\). The key is recognizing that \(v \propto t^{1/2}\) leads to \(s \propto t^{3/2}\).

Answer 1

The Correct Option is D

Step 1: Use the relation between power, force, and velocity 
Power \(P = F \cdot v = ma \cdot v = m \cdot \frac{dv}{dt} \cdot v\) 
Step 2: Rearranging and integrating 
\[ P = m v \frac{dv}{dt} \Rightarrow \frac{P}{m} dt = v dv \] \[ \int v dv = \int \frac{P}{m} dt \Rightarrow \frac{v^2}{2} = \frac{P}{m} t + C \Rightarrow v \propto \sqrt{t} \] Step 3: Use \(v = \frac{ds}{dt}\) 
\[ \frac{ds}{dt} \propto t^{1/2} \Rightarrow ds \propto t^{1/2} dt \Rightarrow \int ds \propto \int t^{1/2} dt \Rightarrow s \propto t^{3/2} \] Step 4: Identify the exponent \(x\) 
So, \(x = \frac{3}{2}\)