Question

The square of the speed of an object undergoing a uniform acceleration a is some function of a and the displacement s, according to the expression

Velocity squared = k*acceleration power m*Speed power n

What dimensions should k have in order for the expression to be dimensionally consistent ?

Answer 1

Velocity squared is measured in $\frac{m^2}{s^2}$ , acceleration - in $\frac{m}{s^2}$ , so acceleration to power m - in $\frac{m^m}{s^{2m}}$ , speed - in $\frac{m}{s}$ , so speed to power n - in $\frac{m^n}{s^n}$ . So, speaking in dimensions:

$\frac{m^2}{s^2}=k\cdot \frac{m^m}{s^{2m}}\cdot \frac{m^n}{s^n}$

$k=\frac{m^2}{s^2}:\frac{m^{m+n}}{s^{2m+n}}$

Answer. The dimension of k is

$k=\frac{m^{2-m-n}}{s^{2-2m-n}}$