Equations of Motion: Quick Revision

A brief note on Equations of Motion. Read it before a test. Try each question at the end before you open its answer.

The big ideas

• For steady acceleration: $v = u + at$, $s = ut + \tfrac12 at^2$, $v^2 = u^2 + 2as$.
• Choose the equation with your three known quantities and the one you want.
• Free fall: without air, everything falls with $g = 9.8\ \text{m/s}^2$.
• Thrown up at $u$: time to the top $\dfrac{u}{g}$, greatest height $\dfrac{u^2}{2g}$; coming down takes as long.
• Circular motion at a steady speed is accelerated: $v = \dfrac{2\pi r}{T}$, velocity along the tangent.

The facts at a glance

A worked example

A driver at $72\ \text{km/h}$ takes $0.8\ \text{s}$ to react, then the car slows at $6\ \text{m/s}^2$. How far does it go before it stops?

1. Units. $72\ \text{km/h} = 20\ \text{m/s}$.
2. Thinking distance. $20\ \text{m/s} \times 0.8\ \text{s} = 16\ \text{m}$. Braking distance. $\dfrac{(20\ \text{m/s})^2}{2 \times 6\ \text{m/s}^2} = 33.3\ \text{m}$.
3. In all. $16\ \text{m} + 33.3\ \text{m} = 49.3\ \text{m}$.

Practice with new numbers

A program to try

The program makes a table of stopping distances.

Mistakes to avoid

• Using the equations when the acceleration changes. Split the motion into parts.
• Forgetting the sign. Slowing down needs a negative $a$; up and down need a chosen positive direction.
• The half in $\tfrac12 at^2$. Leave it out and the distance doubles.
• $g$ at the top of a throw. The velocity is zero there, not the acceleration.
• 'Steady speed round a bend means no acceleration.' The direction changes, so it accelerates.

Check yourself

1. A car speeds up from $4\ \text{m/s}$ to $16\ \text{m/s}$ at $2\ \text{m/s}^2$. How long does it take?

2. A stone is dropped. How far does it fall in $2\ \text{s}$? ($g = 9.8\ \text{m/s}^2$)

3. A ball is thrown up at $9.8\ \text{m/s}$. How long until it reaches the top?

4. Which equation finds a stopping distance when the time is not given?

5. A merry-go-round seat is $4\ \text{m}$ from the centre and goes round once every $8\ \text{s}$. How fast does it move?