Question

Derive the formula for kinetic energy of a particle having mass m and velocity v using dimensional analysis

Answer 1

Kinetic energy of a body depends upon mass (m) and velocity (v) of the body.

Let K.E. ∝ m^x v^y

∴ K.E. = km^x v^y …….. (1)

where,

k = dimensionless constant of proportionality. Taking dimensions on both sides of equation (1),

[L²M¹T^-2] – [L⁰M¹T⁰]^x [L¹M⁰T^-1]^y

= [L⁰M^xT⁰] [L^yM⁰T^-y]

= [L^0+yM^x+0T^0-y]

[L²M¹T²] = [L^yM^xT^-y] …………. (2)

Equating dimensions of L, M, T on both sides of equation (2),

x = 1 and y = 2 ,

Substituting x, y in equation (1), we have

K.E. = kmv²