Acceleration and Acceleration-Time Graphs
Acceleration is the rate of change of velocity with time, a = Δv ÷ Δt, measured in meters per second squared (m/s2), and an acceleration-time graph plots acceleration against time, with the area under the curve equal to the change in velocity. In brief: an object accelerates when it speeds up, slows down or changes direction. Positive acceleration in the positive direction speeds the object up, and acceleration opposite to the velocity slows it down (deceleration). Acceleration is a vector.
When you press the accelerator pedal, brake or turn the steering wheel, you change the velocity of a car, and so you accelerate it. This guide defines acceleration, explains how its sign relates to the motion, shows how to read acceleration-time graphs, compares typical accelerations in a table, works through examples and corrects common mistakes.
What is acceleration?
Average acceleration = change in velocity ÷ time taken:
a = Δv ÷ Δt = (v − u) ÷ t,
where u is the initial velocity and v the final velocity. The unit m/s2 means “meters per second, per second”: an acceleration of 3 m/s2 increases the velocity by 3 m/s every second. The instantaneous acceleration is the slope of the tangent on a velocity-time graph.
Acceleration occurs when:
- the speed increases (a car pulling away from a stop);
- the speed decreases (braking, often called deceleration);
- the direction changes, even at constant speed (a car going around a bend). In circular motion the acceleration points toward the center.
How does the sign of acceleration relate to motion?
In one dimension, the signs of velocity and acceleration together give the motion:
- Same sign (both positive or both negative): the object speeds up.
- Opposite signs: the object slows down.
- Acceleration zero: velocity is constant.
A negative acceleration does not always mean slowing down: if the velocity is also negative, the object speeds up in the negative direction. An object can have zero velocity and nonzero acceleration at the same moment, as a ball does at the top of its path when thrown upward (a = −9.8 m/s2, v = 0).
What does an acceleration-time graph show?
The graph has time on the horizontal axis and acceleration on the vertical axis.
- A horizontal line at zero shows constant velocity (or rest).
- A horizontal line above or below zero shows constant acceleration.
- The area between the line and the time axis is the change in velocity: Δv = a × t for constant acceleration. Area above the axis is positive and area below is negative.
- A sloping line shows that the acceleration itself is changing (this rate of change is called jerk).
Typical accelerations
| Situation | Acceleration (m/s2) |
|---|---|
| Comfortable lift (elevator) | about 1 |
| Car pulling away from a stop | about 2 to 4 |
| Free fall near Earth’s surface | 9.8 |
| Car braking hard on a dry road | about 7 to 9 |
| Sprinter at the start | about 4 to 5 |
Worked example: calculating acceleration and change in velocity
- Speeding up: a car goes from 0 to 27 m/s (about 97 km/h) in 6.0 s. a = (27 − 0) ÷ 6.0 = 4.5 m/s2.
- Slowing down: a train moving at 20 m/s brakes to rest in 5.0 s. a = (0 − 20) ÷ 5.0 = −4.0 m/s2. The negative sign shows that the acceleration is opposite to the motion.
- From an acceleration-time graph: a cart has an initial velocity of 3 m/s. Its acceleration is +2 m/s2 from 0 to 5 s, zero from 5 s to 8 s, and −1 m/s2 from 8 s to 12 s.
- Area from 0 to 5 s: 2 × 5 = +10 m/s.
- Area from 5 to 8 s: 0.
- Area from 8 to 12 s: −1 × 4 = −4 m/s.
- Total change in velocity = 10 + 0 − 4 = +6 m/s. The final velocity is 3 + 6 = 9 m/s.
How was acceleration studied?
Galileo found that balls rolling down an incline gain speed at a steady rate, and that falling bodies accelerate equally, whatever their masses, if air resistance is ignored. Newton showed in 1687 that acceleration is caused by a net force (a = F ÷ m). Accelerometers are now used in phones, cars and aircraft to measure acceleration.
Which mistakes are common?
Mistake 1: Confusing velocity with acceleration
A high velocity does not mean a high acceleration. A plane cruising at 250 m/s has zero acceleration if the velocity is constant.
Mistake 2: Thinking that negative acceleration always means slowing down
It depends on the direction of the velocity as well.
Mistake 3: Forgetting to square the time in the unit
The unit is m/s2, not m/s.
Mistake 4: Thinking that zero velocity means zero acceleration
At the highest point of a throw, the velocity is zero for an instant but the acceleration is still 9.8 m/s2 downward.