The three equations of motion

Three equations for steady acceleration

When the acceleration stays the same, three short equations answer almost every question about the motion:

Here $u$ is the starting velocity, $v$ the final velocity, $a$ the acceleration, $t$ the time and $s$ the distance.

Where do they come from?

The first is the meaning of acceleration: the velocity grows by $a$ each second. The second is the area under the velocity-time graph, a trapezium.

The third comes from putting the first two together, so it needs no time.

Which equation should I use?

Write down the three things you know and the one you want. Pick the equation that has exactly those four. If the time is not given or asked, use the third.

You should find that with $a = 0$ the curve becomes a straight line (steady speed), and that with a negative $a$ the curve bends over: the train slows, stops and would then move back.

Watch out. These equations work only when the acceleration is steady. For a car that speeds up, cruises and brakes, use them one part at a time.

A worked example

A train at $5\ \text{m/s}$ speeds up at $0.5\ \text{m/s}^2$ for $20\ \text{s}$. Find its final velocity and the distance it covers.

1. Velocity. $v = 5\ \text{m/s} + 0.5\ \text{m/s}^2 \times 20\ \text{s} = 15\ \text{m/s}$.
2. Distance. $s = 5\ \text{m/s} \times 20\ \text{s} + \tfrac12 \times 0.5\ \text{m/s}^2 \times (20\ \text{s})^2 = 100\ \text{m} + 100\ \text{m} = 200\ \text{m}$.
3. Check. $v^2 = 225\ \text{m}^2/\text{s}^2$ and $u^2 + 2as = 25\ \text{m}^2/\text{s}^2 + 200\ \text{m}^2/\text{s}^2 = 225\ \text{m}^2/\text{s}^2$. They agree.

Practise with new numbers

Try it in Python

The program works out the train and checks the third equation.

Remember these

• $v = u + at$, $s = ut + \tfrac12 at^2$, $v^2 = u^2 + 2as$.
• They hold only for steady acceleration.
• Pick the equation that has the three things you know and the one you want.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A cyclist at rest speeds up at $1.5\ \text{m/s}^2$ for $6\ \text{s}$. What is her final speed?

2. How far did she go in those $6\ \text{s}$?

3. Which equation would you use to find a stopping distance when the time is not given?

4. A car doubles its speed before braking. How much longer is its stopping distance, with the same brakes?