Question

You know that the kinetic energy of a particle of mass m is  1/2mv² while its potential energy is mgh, where v is the velocity of the particle, h is its height from the ground and g is the acceleration due to gravity. Since the two expressions represent the same physical quantity i.e, energy, their dimensions must be the same. Let us prove this by actually writing the dimensions of the two expressions. 

Answer 1

The dimensions of 1/2 mv² are M.(LT^–1)², or ML² T^–2. (Remember that the numerical factors have no dimensions.) The dimensions of mgh are M.LT^–2.L, or Motion, Force and Energy ML² T^–2. Clearly, the two expressions are the same and hence represent the same physical quantity. 

Let us take another example to find an expression for a physical quantity in terms of other quantities.