In Acceleration acceleration was how quickly velocity changes. Now velocities carry signs, so acceleration does too, and its sign holds a surprise.
Key idea
$a = \dfrac{\Delta v}{\Delta t} = \dfrac{v_f - v_i}{\Delta t}$. An object speeds up when $a$ and $v$ point the same way and slows down when they point opposite ways. The sign of $a$ alone doesn't tell you which.
Why it works
Acceleration points in the direction of the change in velocity. A car moving in the $-x$ direction at $-10\ \text{m/s}$ that speeds up to $-16\ \text{m/s}$ has a change of $-6\ \text{m/s}$: its acceleration is negative and it is speeding up. "Negative" is a direction on your axis, not a word for "slowing".
| v | a | The object is… |
|---|---|---|
| + | + | speeding up |
| + | − | slowing down |
| − | − | speeding up |
| − | + | slowing down |
Worked examples
Worked example: Speeding up in the minus direction
A bus heading west speeds up from $10\ \text{m/s}$ to $16\ \text{m/s}$ in $3\ \text{s}$. Take east as $+x$.
- Heading west, both velocities are negative: $v_i = -10\ \text{m/s}$, $v_f = -16\ \text{m/s}$.
- $$a = \dfrac{-16 - (-10)}{3} = \dfrac{-6}{3} = -2\ \text{m/s}^2.$$
- Negative acceleration, and the bus is speeding up, because $a$ and $v$ are both negative.
Answer: $a = -2\ \text{m/s}^2$
Worked example: Through a turnaround
A ball rolls up a ramp at $+6\ \text{m/s}$. Five seconds later it rolls down at $-4\ \text{m/s}$. Find its acceleration.
- $a = \dfrac{-4 - 6}{5} = -2\ \text{m/s}^2$.
- The ball slowed, stopped for an instant and sped up the other way, with the same acceleration the whole time.
Answer: $-2\ \text{m/s}^2$
Common mistake
Thinking a negative acceleration always means slowing down. It means slowing down only when the velocity is positive. Check by comparing directions: draw the $v$ arrow and the $a$ arrow. Same way, speeding up; opposite ways, slowing down. The word "deceleration" is fine in everyday talk, but in a problem always write $a$ with its sign.