Sketch the position-time graph from a velocity-time graph
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?
A delivery drone flies straight down a warehouse aisle at a steady 3 m/s for 4 s, starting 1 m from the doorway. You've already turned positions into velocities. Now run the translation the other way: from the drone's velocity record to where the drone is at every moment.
The drone's flight on one time axis: how fast on top, and beneath it, where the drone is. The lower graph is the one you are about to sketch.
Start the position-time graph at the given starting position. The drone's first point is x₀ = 1 m at t = 0.
Each flat level moves the drone by that stretch's displacement: the level's height times its width. That works because the area between a velocity-time graph and the time axis is the displacement, with area below the axis counting as negative. For the drone's level:
displacement = 3 × 4 = 12 m
ending position = 1 + 12 = 13 m
While the velocity holds one constant value, the position changes by the same amount each second. So the position-time graph over that stretch is one straight line: the drone's runs from 1 m up to 13 m. Its slope is 3 m/s, the velocity itself, because the slope of a position-time graph at a moment is the velocity at that moment, sign included.
For several levels, work stretch by stretch. Carry each stretch's ending position in as the next stretch's starting position. The line never jumps.
Where the velocity is zero, the drone hovering, the position does not change. Draw a flat line at the position the drone already has.
Where the velocity is negative, the drone heading back toward the doorway, the position falls. The line slopes downward, and it may cross into negative positions.
A lab cart's velocity-time graph: a flat level at −2 m/s from 0 s to 3 s. The cart starts at position 5 m.
Sketch the cart's position-time graph on paper. Mark the starting and ending positions. Then continue to compare your sketch with the model answer.
The model answer. Check your sketch: the displacement is −2 × 3 = −6 m, so the ending position is 5 + (−6) = −1 m; a constant velocity draws one straight line, from (0 s, 5 m) down to (3 s, −1 m), crossing the time axis on the way.
A cargo cart starts at position 1 m. The figure shows the cart's velocity-time graph and four candidate position-time graphs, A to D. Which panel shows the correct position-time graph?
A parade float's velocity-time graph: 2 m/s from 0 s to 3 s, then 0 m/s from 3 s to 5 s. The float starts at position 2 m.
Sketch the float's position-time graph on paper. Then continue to compare your sketch with the model answer.
The model answer. Check your sketch: the first stretch's displacement is 2 × 3 = 6 m, so the line rises from 2 m to 8 m by 3 s; the second stretch has zero velocity, so the line holds flat at 8 m to 5 s.
A river boat's velocity-time graph: one flat level. The boat starts at position 4 m.
On paper, sketch the boat's position-time graph for the stretch shown. Label both axes with quantity and unit, and mark the starting and ending positions. Then continue to compare your sketch with the model answer.
Check your sketch: the line starts at 4 m at t = 0, the given starting position, not zero; the displacement is 2 × 3 = 6 m, so the line ends at 4 + 6 = 10 m at 3 s; a constant velocity draws one straight line, with slope 2 m/s.
A delivery robot's velocity-time graph: one flat level below the time axis. The robot starts at position 4 m.
On paper, sketch the robot's position-time graph for the stretch shown. Label both axes with quantity and unit, and mark the starting and ending positions. Then continue to compare your sketch with the model answer.
Check your sketch: the line starts at 4 m at t = 0; the displacement is −3 × 2 = −6 m, sign kept, so the line ends at 4 + (−6) = −2 m at 2 s; the line falls as one straight line and crosses the time axis rather than stopping at it.
Your turn
A car's velocity-time graph with three flat levels. The car starts at position 2 m.
On paper, sketch the car's position-time graph for the full stretch shown. Label both axes with quantity and unit, and mark the position at 0 s, 2 s, 4 s, and 7 s. Then continue to compare your sketch with the model answer.
Check your sketch: carry the ending position forward, 2 + 6 = 8 m, then 8 m, then 8 − 6 = 2 m; each constant-velocity stretch draws one straight line whose slope is that stretch's velocity, sign included; zero velocity holds the position, and negative velocity brings it back down.
A survey cart starts at position 3 m. The figure shows the cart's velocity-time graph and four candidate position-time graphs, A to D. Which panel shows the correct position-time graph?
A warehouse shuttle's velocity-time graph with three flat levels. The shuttle starts at position 3 m.
On paper, sketch the shuttle's position-time graph for the full stretch shown. Label both axes with quantity and unit, and mark the position at 0 s, 3 s, 4 s, and 6 s. Then continue to compare your sketch with the model answer.
Check your sketch: carry the ending position forward, 3 − 6 = −3 m, then −3 m, then −3 + 8 = 5 m; each level's displacement is height × width, sign included; the slopes of your three lines should be −2 m/s, 0 m/s, and 4 m/s, the velocities themselves.