An automobile of mass 'm' accelerates starting from origin and initially at rest, while the engine supplies constant power P. The position is given as a function of time by :

Solution

Physics sir · 4 steps

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

1. Energy from power. The car starts from rest and all the engine's work goes into kinetic energy. With constant P, the work done in time t is Pt, so Pt = 1/2mv².
2. Find the speed. Solving gives v = √(2P/m) t^(1/2). The speed grows as √t, not linearly, because the force falls as the car speeds up.

   x = ∫₀ᵗ √(2P/m) t^(1/2) dt = √(2P/m) × 2/3 t^(3/2)

3. Tidy the constant. Move 2/3 inside the root as 4/9: 2P/m × 4/9 = 8P/9m. So x = (8P/9m)^(1/2) t^(3/2).
4. So. The position is (8P/9m)^(1/2) t^(3/2), which is option (d).

The idea to keep. For constant power from rest, use Pt = 1/2mv². It gives v ∝ t^(1/2) and x ∝ t^(3/2). Remember these two powers of t.

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