A toy train is moving horizontally on a frictionless straight track under the influence of a constant power supplying source. Which of the plot shown below represents the displacement time curve for its motion.

EXPLANATION:

Work done = W = F⋅s

[F = force, s = displacement]

Power P = rate of work done

= dW/dt = \(\frac{ \overrightarrow{F}\cdot \overrightarrow{ds}}{dt} = \overrightarrow{F}\cdot \overrightarrow{v} \) = constant. (∵ P = constant given)

→Now from dimensional analysis

[F][v] = [MLT-2] [LT-1] = [ML2T-3] = constant

so, [L2T-3] = constant (∵ mass in constant)

⇒ L2 ∝ T3 

⇒ L ∝ T3/2 

∴ Displacement d ∝ t3/2

• →option (1) is a straight line. only possible when d ∝ t
• →option (3) is also a straight line parallel to the t-axis. only possible when d = constant. 
• →option (2) and (4) are possible when d ∝ xn (n = any positive fraction)

if n < 1 then the graph looks like option (4) (as shown in fig 1).

and if n > 1 then the graph looks like option (2). (as shown in fig 2)

as here n = 3/2. so it will look like option (2). 

So, the correct answer is option (2).