Position, velocity & acceleration graphs
Set an initial velocity and a constant acceleration to watch an object move while its position–time and velocity–time graphs update. The simulator uses x(t) = v₀t + ½at² and v(t) = v₀ + at: velocity is the slope of the position graph, and acceleration is the slope of the velocity graph.
What if you changed…
Give the object positive velocity and negative acceleration. When does it turn around?
See the calculation
- At t = 1.5 s: x = 2 × t + ½ × 1 × t² = 4.13 m.
- v = 2 + 1 × 1.5 = 3.5 m/s.
- End displacement at 6 s = 30 m.
What this model assumes
One-dimensional motion starting at x = 0 with constant acceleration.
Keep an eye on this
Negative acceleration does not always mean slowing down. Compare the signs of velocity and acceleration.
Understand the physics ↗Formulas used in this simulation
- Velocity at time t
- v = v₀ + at
- The v–t graph has slope a
- Position at time t
- x = x₀ + v₀t + ½at²
- This lab starts at x₀ = 0
- Velocity without time
- v² = v₀² + 2a(x − x₀)
- Use when the time is unknown
- Average velocity
- v̄ = Δx/Δt = (v₀ + v)/2
- Second form needs constant a
- Displacement from a v–t graph
- Δx = area under the v–t graph
- Area below the axis counts as negative
- Turnaround time
- t = −v₀/a
- When v₀ and a have opposite signs
Position, velocity & acceleration: common questions
Wondering about the why? Start here.
How do you get velocity from position?
Velocity is the rate of change of position: the slope of the position–time graph, or the derivative v = dx/dt. For x = v₀t + ½at², the derivative is v = v₀ + at.
How do you get position from velocity?
The change in position equals the area under the velocity–time graph. For constant acceleration that area is v₀t + ½at², so x = x₀ + v₀t + ½at².
What does a velocity–time graph tell you?
Its height is the velocity, its slope is the acceleration, and the area between the line and the time axis is the displacement.
What does the slope of a position graph mean?
The slope is velocity. A steeper upward slope means a larger positive velocity; a downward slope means negative velocity.
Is displacement the same as distance?
Displacement is final position minus initial position. Distance is the total length traveled, so reversing direction makes the two different.