The equations of uniformly accelerated motion

Three equations from one graph

A jet starts from rest at the end of a runway and speeds up steadily. Pilots need answers like: how long until take-off speed? How much runway? For a steady (uniform) acceleration, three short equations answer all such questions. They all come from the straight-line velocity-time graph.

1. From the slope. The slope is $a = \dfrac{v - u}{t}$, so

2. From the area. The area under the line is a rectangle $ut$ plus a triangle $\tfrac12 \times t \times (v - u) = \tfrac12 at^2$, so

3. Without the time. Put $t = \dfrac{v - u}{a}$ into the area written as the average velocity times the time, $s = \dfrac{u + v}{2}\,t$. The $t$ disappears:

Here $u$ is the starting velocity, $v$ the velocity after a time $t$, $a$ the steady acceleration and $s$ the displacement in that time.

See them agree

Which equation?

Each equation leaves one quantity out. List what you know and what you want, and pick the equation that leaves out the one you neither know nor want.

Worked examples

A jet on a runway. It starts from rest and speeds up at $4\ \text{m/s}^2$. Take-off speed is $80\ \text{m/s}$.

1. Time. $v = u + at$: $80\ \text{m/s} = 0 + 4\ \text{m/s}^2 \times t$, so $t = 20\ \text{s}$.
2. Runway. $v^2 = u^2 + 2as$: $(80\ \text{m/s})^2 = 0 + 2 \times 4\ \text{m/s}^2 \times s$, so $s = 800\ \text{m}$.

A cart pushed up a ramp. It leaves your hands at $3\ \text{m/s}$ and slows at $1.5\ \text{m/s}^2$. Take up the ramp as positive, so $a = -1.5\ \text{m/s}^2$.

1. When does it stop? $0 = 3\ \text{m/s} - 1.5\ \text{m/s}^2 \times t$, so $t = 2\ \text{s}$.
2. How far up? $0 = (3\ \text{m/s})^2 - 2 \times 1.5\ \text{m/s}^2 \times s$, so $s = 3\ \text{m}$.

Watch out: signs. Choose a positive direction once, and give every vector its sign: $u$, $v$, $a$ and $s$. A slowing body has an acceleration opposite to its velocity. Writing $a = +1.5\ \text{m/s}^2$ for the cart would make it speed up the ramp forever.

Watch out: steady only. These equations are true only when the acceleration is constant. When it changes, we need calculus (Course 2.3).

Practise with new numbers

Try it in Python

The program makes a table for the jet every $5\ \text{s}$, and then finds the runway length from the third equation.

Remember these

• $v = u + at$, $\quad s = ut + \tfrac12 at^2$, $\quad v^2 = u^2 + 2as$.
• Pick the equation that leaves out what you neither know nor want.
• One sign convention for every vector. Steady acceleration only.

Check yourself

1. A scooter starts from rest and speeds up at $3\ \text{m/s}^2$ for $8\ \text{s}$. Find its final velocity and the distance it covers.

2. A car slows steadily from $25\ \text{m/s}$ to $10\ \text{m/s}$ over $105\ \text{m}$. Find its acceleration.

3. You know $u$, $a$ and $s$, and you want $v$. Which equation do you use? And if you want $t$ instead?

4. A car at $12\ \text{m/s}$ brakes at $3\ \text{m/s}^2$. A student writes $a = +3\ \text{m/s}^2$ and gets a stopping distance of $-24\ \text{m}$. Fix the mistake.