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 .

increasing transnational kinetic energy. because the transnational kinetic energy formula consists of only two variables, mass and velocity, increasing one of those properties is the only way to increase an object's transnational kinetic energy. increases to mass and velocity, however, do not have the same impact.