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?

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.