Acceleration and the Equations of Motion: Quick Revision

A brief note on Acceleration and the Equations of Motion. Try each question at the end before you open its answer.

The big ideas

• Acceleration is the rate of change of velocity: $a = \dfrac{v - u}{t}$, in $\text{m/s}^2$.
• Speeding up or slowing down? Same signs of $v$ and $a$: speeding up. Opposite signs: slowing down.
• Velocity-time graph: the slope is the acceleration; the area under the line is the displacement (negative below the axis).
• For a steady acceleration, three equations answer almost every question.

The formulas at a glance

And two results built from them:

The first is the stopping distance (thinking plus braking); the second, the distance in the $n$th second from rest.

A worked example

A train comes into a station at $22\ \text{m/s}$ and brakes steadily, stopping in $242\ \text{m}$.

1. Deceleration. $0 = (22\ \text{m/s})^2 + 2a \times 242\ \text{m}$, so $a = -\dfrac{484\ \text{m}^2/\text{s}^2}{484\ \text{m}} = -1.0\ \text{m/s}^2$.
2. Time. $0 = 22\ \text{m/s} - 1.0\ \text{m/s}^2 \times t$, so $t = 22\ \text{s}$.
3. Check. Average velocity $11\ \text{m/s}$ for $22\ \text{s}$ gives $242\ \text{m}$.

Practice with new numbers

A program to try

Give it $u$, $a$ and $s$; it finds $v$ and $t$ from the equations.

Mistakes to avoid

• "Negative acceleration means slowing down." Only when the velocity is positive. A car reversing faster has a negative acceleration too.
• Signs. Choose a positive direction once, and give $u$, $v$, $a$ and $s$ their signs.
• Forgetting the $\tfrac12$ in $\tfrac12 at^2$, or the 2 in $2as$.
• Braking distance and speed. Double the speed, four times the braking distance.
• "In the 5th second" is one second, not five.

Check yourself

1. A cyclist slows from $9\ \text{m/s}$ to $3\ \text{m/s}$ in $4\ \text{s}$. Find her acceleration.

2. From rest, a cart accelerates at $0.8\ \text{m/s}^2$ for $10\ \text{s}$. How far does it go?

3. A velocity-time graph rises in a straight line from $0\ \text{m/s}$ to $14\ \text{m/s}$ in $7\ \text{s}$. What are the acceleration and the distance?

4. With the same brakes, a car's speed goes from $10\ \text{m/s}$ to $15\ \text{m/s}$. How many times longer is its braking distance?

5. From rest at $2\ \text{m/s}^2$, how far does a body go in the $3$rd second?