Worked example: homogeneity
Physical Quantities & Units - OCR A-Level Physics
Setup
The kinematic equation $v^{2} = u^{2} + 2as$ relates final velocity $v$, initial velocity $u$, acceleration $a$, and displacement $s$. To show it is homogeneousAn equation is homogeneous when every term shares the same SI base units. A standard OCR test of dimensional consistency., we write the base units of each of the three terms and check they are identical.
Worked Example [3 marks]
Show that the equation $v^{2} = u^{2} + 2as$ is homogeneous.
Show Solution
1
LHS
$[v^{2}] = (\text{m s}^{-1})^{2} = \text{m}^{2} \text{ s}^{-2}$.
[1]2
RHS first term
$[u^{2}] = (\text{m s}^{-1})^{2} = \text{m}^{2} \text{ s}^{-2}$.
[1]3
RHS second term
$[2as] = (\text{m s}^{-2})(\text{m}) = \text{m}^{2} \text{ s}^{-2}$. The factor $2$ is a pure number and contributes no units.
[1]Answer
All three terms have base units $\text{m}^{2} \text{ s}^{-2}$, so the equation is homogeneous.
Common Mistake
MEDIUM
Wrong: Writing $[v^{2}] = \text{m s}^{-2}$ by mistakenly subtracting the powers, or dropping the square on $u^{2}$.
Right: $v^{2}$ means $v \times v$, so the units are squared too: $(\text{m s}^{-1})^{2} = \text{m}^{2}\text{ s}^{-2}$. Apply the index to every part of the unit.
Right: $v^{2}$ means $v \times v$, so the units are squared too: $(\text{m s}^{-1})^{2} = \text{m}^{2}\text{ s}^{-2}$. Apply the index to every part of the unit.
Examiner Tips and Tricks
- The mark scheme typically allocates one mark per term checked. Always set out three lines for a three-term equation.
- End with a short conclusion sentence such as "all terms share base units $\text{m}^{2}\text{ s}^{-2}$, so the equation is homogeneous". This earns the final mark.