A body starts from rest with uniform acceleration and acquires a velocity V in time T. The instantaneous kinetic energy of the body after any time t is proportional to:
(a) $\left( {V/T} \right)t$
(b) $\left( {{V^2}/T} \right){t^2}$
(c) $\left( {{V^2}/{T^2}} \right)t$
(d) $\left( {{V^2}/{T^2}} \right){t^2}$

Hint:In order to calculate the instantaneous kinetic energy of the body after any time t, we need to use the laws of motion (uniform acceleration) in this question.

Step by step solution
The formulae for the laws of motions (uniform acceleration) are,
\[v = u + at\],
\[s = \dfrac{1}{2}\left( {u + v} \right)t\],
\[s = ut + \dfrac{1}{2}a{t^2}\] and
\[{v^2} = {u^2} + 2as\].
From the question we can observe that,
$u = 0$
So, $V = 0 + at$
Therefore,
\[a = V/T\]
\[KE = 0.5m{v^2}\]
\[v = at = V{\text{ }}t/T\]
\[K{E_t} = 0.5m{(V/T)^2}\]
Hence, \[KE \propto \dfrac{{{V^2}t{^2}}}{{{T^2}}}\]

Hence, the correct answer to the question is option (d).

Additional Information:In case of uniform acceleration, there are three equations of motion which are also known as the laws of constant acceleration. Hence, these equations are used to derive the components like displacement(s), velocity (initial and final), time(t) and acceleration(a). Therefore, they can only be applied when acceleration is constant and motion is a straight line. The kinetic energy (KE) of an object is the energy that it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes.

Note:While solving this question, one should have to know the laws of motion (uniform acceleration) and also its application. The formula used to solve this question is on laws of motions (uniform acceleration). All the formulas are stated above for reference.