A particle is moving unidirectionally on a horizontal plane under the action of a constant power supplying energy source. The displacement (s) - time (t) graph that describes the motion of the particle is (graphs are drawn schematically and are not to scale) :

Solution

Physics sir · 3 steps

Constant power means the kinetic energy grows linearly with time, so find v from that and integrate.

1. Energy from power. The source supplies energy at a steady rate P. Take the particle to start from rest. Then all of this work goes into kinetic energy, so 1/2mv² = Pt.

   v = √(2P/m) t^(1/2) ⇒ s = ∫₀ᵗ v dt = 2/3√(2P/m) t^(3/2)

2. Shape of the curve. With s ∝ t^(3/2), the curve starts at the origin. The slope v keeps increasing, so the graph bends upwards (concave up), but less steeply than a parabola.

   Graph	What it means	Fits constant power?
   (a) horizontal line	particle at rest, v = 0	no, the energy supplied would have to go somewhere
   (b) curving upwards from origin	speed increasing, s ∝ t^(3/2)	yes
   (c) curving downwards	speed decreasing	no, positive power cannot reduce kinetic energy
   (d) straight line	constant v, zero net force	no, then P = Fv = 0

3. So. The displacement grows as t^(3/2), a curve bending upwards from the origin. That is graph (b).

The idea to keep. For constant power, write Pt = 1/2mv². This gives v ∝ t^(1/2) and s ∝ t^(3/2), and also v ∝ s^(1/3). Remember this chain for every constant-power question.

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