Question

A body is moved along a straight line by a machine delivering constant power. The distance moved by the body in time \(t\) is proportional to

• A. \(t^{1/2}\)
• B. \(t^{3/4}\)
• C. \(t^{3/2}\)
• D. \(t^{2}\)

Hint:

Constant power means force times velocity is constant, so integrate twice to relate velocity and distance to time.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
Power \(P=Fv=mv\frac{dv}{dt}\) is constant. The body is assumed to start from rest.

Step 2: Find the velocity
\(mv\,dv=P\,dt\). Integrating, \(\frac{mv^{2}}{2}=Pt\), so \(v=\sqrt{\frac{2P}{m}}\,t^{1/2}\), that is \(v\propto t^{1/2}\).

Step 3: Find the distance
\(v=\frac{dx}{dt}\propto t^{1/2}\). Integrating again, \(x\propto t^{3/2}\).

Step 4: Compare the options
The distance is proportional to \(t^{3/2}\), which is option (C).

Final Answer:
The distance moved is proportional to \(t^{3/2}\). \[ \boxed{\text{(C) }t^{3/2}} \]