Motion
Equations of uniform acceleration
The four SUVAT equations
| Variable | Symbol | Unit |
|---|---|---|
| Displacement | \( s \) | m |
| Initial velocity | \( u \) | m/s |
| Final velocity | \( v \) | m/s |
| Acceleration | \( a \) | m/s² |
| Time | \( t \) | s |
How to choose the right equation
- List the five variables: \( s,\; u,\; v,\; a,\; t \).
- Write down what you know (given or implied — e.g. "starts from rest" means \( u = 0 \)).
- Identify the unknown you are solving for.
- Pick the equation that contains your three knowns and the one unknown (no extra unknowns).
- Substitute and solve, keeping units consistent.
Example 1 — find final velocity (no distance given)
A train accelerates uniformly from 5 m/s at 2 m/s² for 8 s. Find the final velocity.
Known: \( u = 5\,\text{m/s},\quad a = 2\,\text{m/s}^2,\quad t = 8\,\text{s} \). Unknown: \( v \).
Use equation (1): \( v = u + at = 5 + (2)(8) = 5 + 16 = 21\,\text{m/s} \)
Example 2 — find displacement (no final velocity given)
A ball rolls from rest and accelerates at 3 m/s² for 4 s. How far does it travel?
Known: \( u = 0,\quad a = 3\,\text{m/s}^2,\quad t = 4\,\text{s} \). Unknown: \( s \).
Use equation (3): \( s = ut + \tfrac{1}{2}at^2 = (0)(4) + \tfrac{1}{2}(3)(4)^2 = 0 + 24 = 24\,\text{m} \)
Example 3 — no time given (use equation 4)
A car brakes from 30 m/s to rest with deceleration 6 m/s². Find the braking distance.
Known: \( u = 30,\quad v = 0,\quad a = -6\,\text{m/s}^2 \). Unknown: \( s \).
Use equation (4): \( v^2 = u^2 + 2as \)
\( 0 = 30^2 + 2(-6)s \Rightarrow 0 = 900 - 12s \Rightarrow s = \dfrac{900}{12} = 75\,\text{m} \)
Sign convention
- Motion in the positive direction: \( s > 0,\quad v > 0,\quad a > 0 \) (if accelerating).
- Deceleration means \( a \) is opposite to the direction of motion — write it as negative.
- "Comes to rest" means \( v = 0 \).
- "Starts from rest" means \( u = 0 \).
- "Falls freely" or "thrown upward" — take downward as positive (or upward — be consistent).