Question

Let us consider an equation : 1/2 mv² = mgh, where m is the mass of the body, v its velocity, g the acceleration due to gravity and h is the height. check whether this equation is dimensionally correct.

Answer 1

The dimensions of L.H.S. are

[M] [LT^-1]² = [L²T^-2] = [ML²T^-2]

The dimensions of R.H.S. are

[M] [LT^-2] [L] = [M] [L²T^-2] = [ML²T^-2]

The dimensions of L.H.S. and R.H.S. are same, hence the equation is dimensionally correct.

Answer 2

In order to compare the dimensions, the constant terms are ignored.

mu² = mgh

Substitute the units for  both the quantities:

⇒ kg(ms⁻¹)² = kgms⁻²m

Compare the dimensions:

⇒ [ML²T⁻²] = [ML²T⁻²]

The equation is dimensionally correct.