Which division gives a trip's average velocity?
A boat crosses a calm river at a steady 5 m/s and holds that velocity for 6 s. How far does it travel? You've already seen the equation to use. Average velocity is v = Δx/Δt, and while the velocity is constant, rearranging it gives the displacement: Δx = v × Δt.
A boat crosses a calm river at a steady 5 m/s and holds that velocity for 6 s. How far does it travel?
Write down the values in the question:
v = 5 m/s
Δt = 6 s
Write down the equation:
Δx = v × Δt
Substitute in the values:
Δx = (5)(6)
Simplify:
Δx = 30 m
Now look at the same crossing as a velocity-time graph.
The shaded region under the line is a rectangle. Its height is the velocity, 5 m/s. Its width is the time, 6 s.
The area of a rectangle is height × width: exactly the multiplication you just did. Same numbers, same product. The rectangle's area is the boat's displacement, 30 m. And the units agree too: m/s × s gives m.
A cart holds a constant velocity. Why does the rectangle under its velocity-time line equal the cart's displacement?
If the boat reversed, its velocity would be negative, so v × Δt would come out negative. And its rectangle would hang below the axis. That is why area below the axis counts as negative displacement.
Why does area below the time axis count as NEGATIVE displacement?
But what if the velocity is changing, so the line curves and no single rectangle fits?
Cut the time into thin slices. In a thin enough slice, the velocity barely changes, so that slice's area is that slice's displacement.
Add up all the slices. The total area is the total displacement, whatever shape the line is. That is why the area between a velocity-time graph and the time axis gives the displacement, for any motion.
A bus's velocity-time line is curved, never flat. Why can the area under it still give the bus's displacement?
Another boat's velocity-time line is flat. Two equal rectangles are shaded beneath it: P from t = 0 to 3 s, and Q from t = 3 s to 6 s. What do the equal areas of P and Q say about the motion?
A student says: 'For a curved velocity-time line, the area rule is only approximate, because the velocity inside each slice is never exactly constant.' What is the best response?
A cart moves with a constant negative velocity for a while. Its velocity-time line is flat, below the time axis, with the rectangle between line and axis shaded. Why is the cart's displacement negative over that interval?