A body at rest is moved along a horizontal straight line by a machine delivering a constant power. The distance moved by the body in time 't' is proportional to :

Solution

Physics sir · 3 steps

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

1. Energy from power. The machine delivers power P at a constant rate, so the work done in time t is Pt. The body starts from rest and moves on a horizontal line, so all of this work becomes kinetic energy.

   Pt = 1/2mv² ⇒ v = √(2P/m) t^(1/2)

2. Integrate for distance. Distance is the integral of speed: s = ∫₀ᵗ √(2P/m) t^(1/2) dt = 2/3√(2P/m) t^(3/2).
3. So. Everything in front of t^(3/2) is constant, so s ∝ t^(3/2). The answer is (a).

The idea to keep. For constant power from rest: KE ∝ t, so v ∝ t^(1/2) and s ∝ t^(3/2). Each integration with respect to time adds 1 to the power of t.

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