A car pulling away from a stop light, a bike braking, a skateboarder carving a turn: in each, the velocity is changing. How fast it changes is the acceleration. Finding it is a subtraction and a division, and the signs come straight from signed-number arithmetic.
Key idea
Acceleration is the change in velocity divided by the time it took: $$a = \dfrac{\Delta v}{\Delta t} = \dfrac{v_f - v_i}{t}$$ Its unit is meters per second per second, written m/s².
Why it works
If a car's velocity goes from 0 to 12 m/s in 4 s, it gains 3 m/s every second: 3, 6, 9, 12. So its acceleration is 3 (m/s) per second, or 3 m/s². The "per second" appears twice because acceleration is a rate of change of a rate of change.
Worked example: Speeding up
A car goes from 4 m/s to 22 m/s in 6 s. Find its acceleration.
- Change in velocity: $\Delta v = 22 - 4 = 18$ m/s.
- $a = \dfrac{18\,\mathrm{m/s}}{6\,\mathrm{s}} = 3\,\mathrm{m/s^2}$.
Answer: $3\,\mathrm{m/s^2}$
Worked example: Slowing down
A bike moving forward at 10 m/s brakes to a stop in 4 s. Take forward as positive. Find the acceleration.
- $\Delta v = 0 - 10 = -10$ m/s.
- $a = \dfrac{-10\,\mathrm{m/s}}{4\,\mathrm{s}} = -2.5\,\mathrm{m/s^2}$.
- The minus sign says the acceleration points backward, against the motion, so the bike slows down.
Answer: $-2.5\,\mathrm{m/s^2}$
Common mistake
Thinking acceleration means "going fast." A jet cruising at a steady 250 m/s in a straight line has zero acceleration; a car just starting from rest has a large one. Slowing down is acceleration too, and so is turning at a steady speed, because the direction of the velocity changes.
Anything dropped near Earth, with air resistance too small to matter, gains about 9.8 m/s of downward speed every second. That is an acceleration of 9.8 m/s², the same for a pebble and a bowling ball.