Question

Dimensions of kinetic energy are the same as that of

• A. Acceleration
• B. Force
• C. Work
• D. Pressure

Answer 1

The correct option is C

Work

Reason for correct option-

1. Kinetic energy is a form of energy that an object possesses by virtue of its motion
2. Therefore, it has the same dimensional formula as energy
3. $K.E=\frac{m{v}^2}{2}$
4. $\left[K.E\right]=\left[m\right]{\left[v\right]}^2=\left[M\right]{\left[\frac{L}{T}\right]}^2=\left[M^1{L}^2{T}^{-2}\right]$
5. Work is the energy transferred to the body by the application of force
6. $Work=force\times dispacement$
7. $force=mass\times acceleration$
8. $acceleration=\frac{velocity}{time}$
9. $$\begin{gathered} \left[W\right]=\left[F\right]\left[L\right]=\left[mass\times acceleration\right]\left[L\right]=\left[M\right]\left[\frac{dv}{dt}\right]\left[L\right]=\left[M\right]\left[v\right]{\left[T\right]}^{-1}\left[L\right] \\ \left[W\right]=\left[M\right]\left[L{T}^{-1}\right]{\left[T\right]}^{-1}\left[L\right] \\ \left[W\right]=\left[M^1{L}^2{T}^{-2}\right] \end{gathered}$$
10. Therefore both of them have the same dimensional formula as energy i.e. $\left[M^1{L}^2{T}^{-2}{K}^0\right]$, Here M implies mass, L implies length, T implies time, and K implies the temperature

Reason for incorrect option

Option A

1. Acceleration is defined as the rate of change of velocity with respect to time
2. $acceleration=\frac{velocity}{time}$
3. Hence its dimensional formula is $\left[M^0{L}^1{T}^{-2}{K}^0\right]$ , which is not equal to kinetic energy

Option B

1. Force applied is the rate of change of momentum of the body.
2. $force=mass\times acceleartion$
3. Hence its dimensional formula is $\left[M^1{L}^1{T}^{-2}{K}^0\right]$ , which is not equal to kinetic energy

Option D

1. Pressure is defined as force per unit area
2. $pressure=\frac{force}{Area}$
3. Thus, its dimensional formula is $\left[M^1{L}^{-1}{T}^{-2}{K}^0\right]$ , which is not equal to kinetic energy

Hence, option C is the correct answer.