Draw the velocity-time graph from an acceleration-time graph
The figure shows a sports car's acceleration-time graph. Calculate the car's change in velocity over the full stretch shown, in m/s.
A car creeps along a highway on-ramp at 2 m/s, waiting for a gap. Then the driver floors it: a steady 3 m/s² for the next 4 s. You've already worked out what one stretch of steady acceleration does to a velocity. Now you draw the car's whole velocity story.
The car's motion told twice on one time axis: its acceleration-time graph on top, and beneath it the velocity-time graph you are learning to draw.
Start the velocity-time graph at the car's starting velocity. The first point is v₀ = 2 m/s at t = 0.
The rectangle on the acceleration-time graph gives the change. For any stretch: ending velocity = starting velocity + the rectangle's area.
area = 3 × 4 = 12, so Δv = 12 m/s
v at 4 s = 2 + 12 = 14 m/s
While the acceleration holds one constant value, the velocity climbs by the same amount each second. So the velocity-time graph over that stretch is one straight line, here from (0 s, 2 m/s) up to (4 s, 14 m/s). The line's steepness is forced too, because the slope of a velocity-time graph at a moment is the acceleration at that moment, sign included.
For each stretch: start from the velocity the object already has, add the rectangle's area to find the ending velocity, and join the two points with one straight line.
Where the acceleration-time graph reads zero, the velocity does not change. Draw a flat line at the velocity the object already has.
A negative acceleration works the same way with the sign included. You've already seen that a negative acceleration shifts the velocity toward the negative direction each second, so the line ramps downward.
A taxi's acceleration record: 2 m/s² held from 0 s to 4 s. The taxi starts at 3 m/s.
Sketch the taxi's velocity-time graph on paper. Mark the starting and ending velocities. Then continue to compare your sketch with the model answer.
The model answer. Check your sketch: the line starts at 3 m/s; the rectangle's area is 2 × 4 = 8, so the velocity changes by 8 m/s; the line ends at 3 + 8 = 11 m/s at 4 s; one constant acceleration draws one straight ramp.
A delivery van's starting velocity is 6 m/s. The figure shows the van's acceleration-time graph and four candidate velocity-time graphs, A to D. Which panel shows the correct velocity-time graph?
For a graph with several rectangles, work stretch by stretch. Carry the ending velocity of each stretch in as the starting velocity of the next.
A light-rail train's acceleration record: 3 m/s² from 0 s to 2 s, then 0 m/s² from 2 s to 5 s. The train starts at 4 m/s.
Sketch the train's velocity-time graph on paper. Then continue to compare your sketch with the model answer.
The model answer. Check your sketch: the first stretch's area is 3 × 2 = 6, so the line ramps from 4 m/s to 10 m/s by 2 s; the second stretch has zero acceleration, so the line holds flat at 10 m/s to 5 s.
A go-kart's acceleration-time graph. The go-kart's starting velocity is 5 m/s.
On paper, draw the go-kart's velocity-time graph for the stretch shown. Label both axes with quantity and unit, and mark the starting and ending velocities. Then continue to compare your drawing with the model answer.
Check your drawing: the graph starts at 5 m/s at t = 0, the given starting velocity, not zero; the line is one straight ramp, not a curve and not a step; it ends at 13 m/s at 4 s, because the rectangle's area, 2 × 4 = 8, is added to the 5 m/s start; the axes are labeled velocity (m/s) and time (s).
A boat's acceleration-time graph: a level below the time axis. The boat's starting velocity is 8 m/s.
On paper, draw the boat's velocity-time graph for the stretch shown. Label both axes with quantity and unit, and mark the starting and ending velocities. Then continue to compare your drawing with the model answer.
Check your drawing: the acceleration is −2 m/s², so the velocity shifts 2 m/s toward the negative direction each second; over 3 s it falls by 6 m/s, from 8 m/s to 2 m/s; one constant acceleration draws one straight line, here ramping downward; the axes are labeled velocity (m/s) and time (s).
Your turn
A bus's acceleration-time graph. The bus's starting velocity is 6 m/s.
On paper, draw the bus's velocity-time graph for the full stretch shown. Label both axes with quantity and unit, and mark the velocity at 0 s, at 2 s, and at 5 s. Then continue to compare your drawing with the model answer.
Check your drawing: the velocity holds at 6 m/s until 2 s, because the acceleration there is zero; then the rectangle's area is 3 × 3 = 9, so the line ramps from 6 m/s to 15 m/s by 5 s; each stretch's ending velocity carries into the next.
A subway train's starting velocity is 10 m/s. The figure shows the train's acceleration-time graph and four candidate velocity-time graphs, A to D. Which panel shows the correct velocity-time graph?
An airport shuttle's acceleration-time graph with three levels. The shuttle's starting velocity is 2 m/s.
On paper, draw the shuttle's velocity-time graph for the full stretch shown. Label both axes with quantity and unit, and mark the velocity at 0 s, 2 s, 4 s, and 6 s. Then continue to compare your drawing with the model answer.
Check your drawing: the first stretch adds 4 × 2 = 8, so the line ramps from 2 m/s to 10 m/s by 2 s; the zero stretch holds the line flat at 10 m/s until 4 s; the last stretch shifts the velocity 2 m/s toward the negative direction each second, ending at 10 − 4 = 6 m/s at 6 s.