The area under an acceleration-time graph: the change in velocity
On a velocity-time graph, what does the area between the line and the time axis over an interval give?
A car is already rolling at 4 m/s as it joins the on-ramp. For the next 5 s the driver holds the acceleration at a steady 2 m/s². How much faster is the car moving when those 5 s are up?
Hold onto the car's numbers: in at 4 m/s, then 2 m/s² held for 5 s. One reading of the car's acceleration record answers the question.
The car's acceleration, recorded moment by moment through the 5 s: a steady 2 m/s² the whole way.
Each second at 2 m/s² adds 2 m/s of velocity. Over 5 s the car climbs from 4 m/s to 14 m/s: a gain of 10 m/s.
Between the car's flat line and the time axis sits a rectangle: height 2 m/s², width 5 s.
Height × width: 2 × 5 = 10. That 10 is exactly the 10 m/s the car gained. The rectangle's area is the change in velocity.
The units agree. Multiplying m/s² by s gives m/s, so the area comes out as a velocity change, not an acceleration.
Why does the area give the velocity change? You've already seen the equation a = Δv/Δt. Multiply both sides by Δt: the change in velocity is a × Δt. And height times width is exactly the rectangle's area.
The area between an acceleration-time graph and the time axis is the change in velocity. This holds for any motion, not just the car's.
The figure shows a different car's acceleration-time graph, with the region between the graph and the time axis shaded. What does the shaded area give?
Careful with what the area is: a change. The area gives the change in the car's velocity, not the velocity itself. This car rolled in at 4 m/s and left at 14 m/s. A car that rolled in at 30 m/s would gain the same 10 m/s and leave at 40 m/s.
The graph shows a subway train's acceleration. What does the shaded rectangle's area tell you about the train?
Read the rectangle off the graph: its height is 4 m/s² and its width is 3 s.
Multiply height by width:
4 × 3 = 12
The area is 12, so the train's velocity changes by 12 m/s, because the area between an acceleration-time graph and the time axis is the change in velocity.
The train's velocity changes by 12 m/s.
Keep the count straight. Only two of the three motion graphs carry an area meaning. The area under a velocity-time graph is the displacement. The area under an acceleration-time graph is the change in velocity. A position-time graph's area means nothing.
The figure shows a motorboat's acceleration-time graph with the region between the graph and the time axis shaded. The shaded area works out to 9. Which reading of that 9 is correct?
Your turn
The figure shows an elevator's acceleration-time graph. A student shades the region between the graph and the time axis and calculates its area. Which statement uses that area correctly?
A velocity-time graph and an acceleration-time graph each enclose an area with the time axis. Which pairing of area and meaning is correct?
The figure shows a delivery van's acceleration-time graph. A student says: 'The area under the graph is 15, so the van ends up moving at 15 m/s.' Why is the student's claim wrong?