A body is moving unidirectionally under the influence of a constant power source. Its displacement in time t is proportional to :

Solution

Physics sir · 3 steps

Constant power means kinetic energy grows linearly with time, so start from 1/2mv² = Pt.

1. Energy from power. Power is the rate of doing work, so in time t the source does work Pt. By the work-energy theorem, this all goes into kinetic energy if the body starts from rest.

   1/2mv² = Pt ⇒ v = √(2P/m) t^(1/2)

2. Integrate velocity. Since v = dx/dt ∝ t^(1/2), integrating gives x ∝ t^(3/2)/(3/2). The motion is in one direction, so displacement equals distance and nothing cancels.
3. So. Displacement is proportional to t^(3/2), which is option (b).

The idea to keep. For constant power, write Pt = 1/2mv². Then v ∝ t^(1/2), x ∝ t^(3/2) and a ∝ t^(-1/2). Under constant power the force falls as the body speeds up.

Answer from the NTA final answer key. Solution by PrepWiser.