Suppose you are given the following equation, where xf and xi represent positions at two instants of time, vxi is a velocity, ax is an acceleration, t is an instant of time, and a, b, and c are integers. xf = xita + vxitb + ½axtc. Required:
For what values of a, b, and c is this equation dimensionally correct?

Explanation:

xf = xita + vxitb + ½axtc.

xf is displacement , dimensional formula L .

Xi initial displacement , dimensional formula L

t is time , dimensional formula T ,

vxi is velocity , dimensional formula LT⁻¹

ax is acceleration , dimensional formula = LT⁻²

xf = xi t a + vxi t b + ½ ax t c.

L = aLT + b LT⁻¹ T + c LT⁻² T

From the law of uniformity , dimensional formula of each term of RHS must be equal to term on LHS

aLT = L

a = T⁻¹

b LT⁻¹ T = L

b = 1 ( constant )

c LT⁻² T = L

c = T

so a = T⁻¹ , b = constant and c = T .