Velocity-Time Graphs
Velocity-Time Graphs
A velocity-time graph does not show position. Its height is velocity. Its gradient shows how velocity changes, and the area between the line and the time axis records the displacement accumulated over time.
The gradient is the acceleration
- Rise over run here is , and that is the definition of acceleration.
- A straight line means constant acceleration. A flat line means constant velocity and zero acceleration. A negative gradient means negative acceleration; whether the object slows down also depends on the sign of its velocity.
- If the v-t line curves, the acceleration is changing. The gradient of a tangent gives the acceleration at that instant, the same tangent method as lesson 2.2.
Worked example
Warm-up: acceleration from a v-t line
A v-t line rises from 0 to 15 m s⁻¹ in 6.0 s. Find the acceleration.
- .
The area is the displacement
- In a short time, displacement = velocity × time. On the graph that product is a thin strip of area under the line. Adding all the strips gives the whole area, so area = displacement.
- Split the area into triangles and rectangles, find each one, and add them with their signs. For a curved line, count squares or use a suitable numerical estimate if the question supplies a scale.
Worked example
The standard two-part journey
An object accelerates from rest at 2.5 m s⁻² for 6.0 s, then travels at constant velocity for 4.0 s. Find the total displacement.
- Velocity reached: .
- Triangle (0 to 6 s): .
- Rectangle (6 to 10 s): .
- Total: .
Drag the sliders below to build your own v-t journey and read the areas:
v–t Graph Explorer
Adjust the sliders. The violet dashed triangle shows acceleration (gradient Δv/Δt), and the shaded area shows displacement.
Gradient = acceleration
2 m s⁻²
Shaded area = displacement
s = 36.0 m
| Speed-time graph | Velocity-time graph |
|---|---|
| height = speed | height = signed velocity |
| area = distance | signed area = displacement |
| cannot go below the time axis | may go below the time axis |
| gradient = rate of change of speed | gradient = acceleration |
Area below the axis is negative
- When the line drops below the time axis, the velocity is negative: the object is moving backward.
- Area below the axis counts as negative displacement. Add the areas with their signs to get the displacement; add their sizes to get the total distance.
Worked example
A harder step: displacement and distance
A v-t line runs from 8.0 m s⁻¹ at t = 0 down through zero at t = 4.0 s to −4.0 m s⁻¹ at t = 6.0 s. Find the displacement and the total distance.
- Area above the axis (0 to 4 s): .
- Area below the axis (4 to 6 s): , counted as .
- Displacement: . Distance: .
One v-t graph answers two different questions. Gradient gives the acceleration; area gives the displacement, with area below the axis negative.
Common mistake
A velocity-time line falls uniformly from +12 m s⁻¹ at 0 s to −4.0 m s⁻¹ at 8.0 s. Determine the acceleration, the time when the object changes direction, the displacement and the total distance travelled.
Show worked answer
- Gradient: .
- Set : the line loses 12 m s⁻¹ at 2.0 m s⁻², so the direction changes at .
- Signed area: .
- Distance: .