Position, velocity & acceleration, explained.
Velocity is the rate of change of position, and acceleration is the rate of change of velocity. A negative acceleration can slow an object traveling in the positive direction, then make it speed up in the negative direction.
The relationship to remember
Key formulas
- 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
Worked example
Problem. A cart moving at +6 m/s has a constant acceleration of −2 m/s². When does it turn around, how far out does it get, and where is it after 6 s?
- It turns around when v = 0: 0 = 6 + (−2)t, so t = 3 s.
- Position at the turnaround: x = 6 × 3 + ½ × (−2) × 3² = 18 − 9 = 9 m.
- After 6 s: x = 6 × 6 + ½ × (−2) × 6² = 36 − 36 = 0 m, and v = 6 + (−2) × 6 = −6 m/s.
- Displacement over the 6 s is 0 m, but the distance travelled is 9 m out plus 9 m back = 18 m.
Answer. The cart turns around at t = 3 s, 9 m from the start, and is back at x = 0 moving at −6 m/s after 6 s.
A quick check
With initial velocity 2 m/s and acceleration 1 m/s², after 4 s the position is 16 m and the velocity is 6 m/s.
Inside the default experiment
- 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.
When this model applies
One-dimensional motion starting at x = 0 with constant acceleration.
Make a prediction. Then test it.
Give the object positive velocity and negative acceleration. When does it turn around?
Try it: Position, velocity & acceleration graphs ↗Frequently asked questions
Wondering about the why? Start here.
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.