Acceleration
How fast does velocity change?
An auto-rickshaw pulls away from a traffic light. Its speedometer climbs: 0, then 2, then 4, then 6 metres per second, one step every second.
Its velocity changes by 2 m/s every second. That rate of change of velocity is the acceleration: 2 metres per second per second, written 2 m/s².
Average acceleration is the change in velocity divided by the time taken: a = Δv / Δt. Let Δt shrink to zero and you get the instantaneous acceleration, a = dv/dt.
Now plot velocity against time. The v–t graph of the auto is a straight line going up. Its slope, rise over run, is the acceleration.
Acceleration has a direction, so it has a sign. Watch two arrows above a car: green for velocity, amber for acceleration.
When the arrows point the same way, the car speeds up. When they point opposite ways, the car slows down, even if a is positive.
So a negative acceleration does not always mean slowing down. A car moving left, in the negative direction, speeds up when a is negative too.
Area under the v–t graph
A train moves at a steady 20 m/s for 10 s. Its v–t graph is a flat line at 20.
In 10 s it covers 20 × 10 = 200 m. Now look at the graph: the rectangle under the line is 20 tall and 10 wide. Its area is also 200.
This is always true. The area under a v–t graph between two times equals the displacement in that time.
Now the train brakes evenly to rest in 8 s. The graph slopes down to zero, and the area under it is a triangle: half × 8 × 20 = 80 m.
What if the velocity goes negative? A ball thrown up comes back down. Area below the time axis counts as negative displacement.
Add the area above the axis, subtract the area below, and you get the displacement. Add both as positive numbers and you get the total distance travelled.
Try it: slope and area of the v–t graph
- The slope of the v–t graph is the acceleration. Compare it with the a–t graph below.
- Green shading is area above the time axis (positive displacement). Orange is below the axis (negative).
- Watch the car turn around exactly when the v–t line crosses zero.
- Turn on the brakes option to see a car that stops instead of reversing.
Quick check
A ball is thrown straight up. At the very top of its flight, what are its velocity and acceleration?
- Velocity zero, acceleration zero
- Velocity zero, acceleration 9.8 m/s² downwards
- Velocity 9.8 m/s downwards, acceleration zero
- Velocity zero, acceleration 9.8 m/s² upwards
Answer
B. At the top the ball stops for an instant, so v = 0. But gravity never switches off: the velocity is still changing from upwards to downwards at 9.8 m/s every second. If the acceleration were zero there, the ball would stay hanging in the air.
Area under v–t
A car moves at 20 m/s for 4 s, then slows down uniformly and stops after another 6 s. How far does it travel in these 10 s?
- 80 m
- 120 m
- 140 m
- 200 m
Answer
C. Area under the v–t graph = rectangle + triangle = (20 × 4) + ½ × 6 × 20 = 80 + 60 = 140 m. Its deceleration while braking is 20 / 6 ≈ 3.3 m/s².
Summary
- Average acceleration:
ā = Δv / Δt = (v₂ − v₁) / (t₂ − t₁) - Instantaneous acceleration:
a = dv/dt - Displacement from a v–t graph:
Δx = area under the v–t graph - SI unit of acceleration:
m/s² (m s⁻²)
- Acceleration is the slope of the v–t graph. A straight v–t line means uniform (constant) acceleration.
- If v and a have the same sign, the object speeds up. If they have opposite signs, it slows down.
- The area under the v–t graph is the displacement. Area below the time axis counts as negative.
- An object can have zero velocity and non-zero acceleration, like a ball at the top of its flight.