A cyclist rides 300 m north, then 100 m south. Which pair states the distance traveled and the displacement?
A harbor boat sets out from its dock at a steady 5 m/s toward the far pier. Three seconds into the trip a horn sounds: the berth is closed. So the boat slows, reverses, and by t = 5 s is heading back at −5 m/s, holding that until t = 8 s. After all that coming and going, one question matters back at the dock: where did the boat end up?
The shading on the graph answers it.
Over any interval, add the signed areas to get the displacement for that interval. The boat's two shaded regions are the same size, one positive and one negative, so they add to zero. The boat's displacement over the whole 8 s is zero: it ends exactly where it started, back at the dock.
On a velocity-time graph, what does the area between the line and the time axis over an interval give?
Equal areas above and below the axis always mean the object is back where it started. Unequal areas mean the larger side won: the object ends on that side of its starting point.
The figure shades two equal areas on a river boat's velocity-time graph: one above the time axis, then one below it. What is the boat's displacement over the whole interval?
Here is a tractor's velocity-time graph, with the region between the line and the time axis shaded from t = 0 to 6 s. The region sits entirely above the axis.
That shaded area gives the tractor's displacement from t = 0 to 6 s. The displacement is positive, because the region sits above the time axis.
Now a rowboat's graph, with the region shaded from t = 2 s to 7 s. This region lies entirely below the axis, so the area counts as negative: the rowboat's displacement over the interval is negative.
In earlier grades, the area under a speed-time graph gave the distance traveled. For a one-way trip in the positive direction that never reverses, that distance and the displacement are the same number. The signed area tells you more: not just how far the boat traveled, but where it ended up.
The figure shades the region between a luggage cart's velocity-time line and the time axis from t = 1 s to 6 s. The region lies entirely below the axis. Which statement is correct?
A jet ski's velocity-time graph over one out-and-back trip shows equal shaded areas above and below the time axis. Where is the jet ski at the end of the interval?
On a delivery cart's velocity-time graph, the shaded area above the time axis is clearly larger than the shaded area below it. Where does the cart end the interval, relative to its starting point?
Two students read the same velocity-time graph. Student P says the area between the line and the time axis gives the displacement. Student Q says the area gives the distance traveled, so it can never count as negative. Who is right?