A body of mass 2 kg is initially at rest. It starts moving unidirectionally under the influence of a source of constant power P. Its displacement in 4 s is 1/3 α² √P m. The value of α will be ______.

Solution

Physics sir · 4 steps

Constant power means work grows linearly with time, so set Pt equal to kinetic energy and find v(t).

1. Work equals kinetic energy. The body starts from rest, and the source delivers work at a steady rate P. After time t the work done is Pt, and all of it becomes kinetic energy.

   Pt = 1/2mv² ⇒ v = √(2Pt/m) = √P t^(1/2) (m = 2 kg)

2. Integrate for displacement. Since v = dx/dt, we get x = ∫₀ᵗ √P t^(1/2) dt = 2/3√P t^(3/2). At t = 4 s, t^(3/2) = 8, so x = 16/3√P.
3. Compare with given form. Match 16/3√P with 1/3α²√P. This gives α² = 16.
4. So. α = 4, so the answer is 4.

The common slip: 16. Stopping at α² = 16 and reporting that gives 16. Take the square root to get α = 4.

The idea to keep. Under constant power, v ∝ t^(1/2) and x ∝ t^(3/2). Start from Pt = 1/2mv², then integrate v to get the displacement.

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