Velocity & Acceleration
What it really means to speed up — and why the area under a velocity graph is a distance.
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The big idea: Acceleration measures how quickly velocity changes. Just as velocity is the slope of a position–time graph, acceleration is the slope of a velocity–time graph — and the area underneath that graph is the distance travelled.
Acceleration is a change in velocity
Press the accelerator and the car's velocity climbs: 0, 10, 20, 30 km/h. Acceleration is how fast that velocity itself is changing — measured in metres per second, per second (m/s²). An acceleration of 3 m/s² means the velocity grows by 3 m/s every single second.
\[ a = \frac{\Delta v}{\Delta t} \]
Acceleration is the change in velocity divided by the time it took.
🎮 Interactive: acceleration & the v–t graph LIVE
Set the acceleration and press Start. The steeper the velocity–time line, the greater the acceleration. The shaded triangle under the line is the distance travelled — watch it grow.
Two equations for motion from rest
Starting from rest with constant acceleration, velocity grows in a straight line and distance grows as a curve (because you're covering ground faster and faster):
\[ v = a\,t \qquad x = \tfrac{1}{2}\,a\,t^2 \]
Velocity from rest, and the distance covered — the ½at² is the triangular area under the v–t line.
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Why area = distance
For steady motion, distance = speed × time — a rectangle's area on a v–t graph. When velocity climbs steadily from zero, that region is a triangle of area \(\tfrac{1}{2} \times \text{base} \times \text{height} = \tfrac{1}{2} \, t \times (a t) = \tfrac{1}{2} a t^2\). The geometry and the formula are the same thing.
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Accelerating can mean slowing down
Acceleration isn't the same as "getting faster." It's any change in velocity. Hitting the brakes is a negative acceleration (a deceleration). Turning a corner at constant speed is also an acceleration, because direction — part of velocity — is changing.
📝 Worked example: A car starts from rest and accelerates at 4 m/s² for 5 s. Find its final velocity and the distance it covers.
- Final velocity: \(v = a t = 4 \times 5 = 20\,\text{m/s}\).
- Distance: \(x = \tfrac{1}{2} a t^2 = \tfrac{1}{2} \times 4 \times 5^2 = \tfrac{1}{2} \times 4 \times 25\).
✓ v = 20 m/s and x = 50 m.
Check your understanding
1. An object accelerates from rest at 2 m/s². How fast is it moving after 6 s?
v = a·t = 2 × 6 = 12 m/s.
2. On a velocity–time graph, what does the AREA under the line represent?
Area under a v–t graph = velocity × time summed up = displacement.
3. A car is slowing down as it approaches a red light. Its acceleration is…
Slowing down means velocity is decreasing, so the acceleration points opposite to the motion — a negative acceleration, or deceleration.
✅ Key takeaways
- Acceleration is the rate of change of velocity: a = Δv / Δt (units m/s²).
- From rest with constant a: v = a·t and x = ½·a·t².
- On a v–t graph, the slope is the acceleration and the area underneath is the distance.
- Acceleration means any change in velocity — including slowing down or changing direction.