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Instantaneous velocity
Beginner
Velocity at one instant, the derivative of position.
v = dx/dtInstantaneous velocity is the limit of Δx/Δt as Δt → 0, the slope of the tangent on an x–t graph, v = dx/dt. A speedometer shows its magnitude.
If x = 3t² then v = 6t, so at t = 2 s the velocity is 12 m/s.
A velocity of zero for an instant (top of a throw) does not mean zero acceleration.
In more detail
Instantaneous velocity is the velocity at a single moment, found as the limit of average velocity over shorter and shorter time intervals. It is the derivative of position with respect to time.
Real-world uses
- A speedometer shows the instantaneous speed of a car at each moment.
- Radar guns measure the instantaneous velocity of a passing vehicle.
- A police radar gun measures the instantaneous speed of a car at the moment it is aimed, not the average over the whole journey.
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Frequently asked questions
How do I find instantaneous velocity from a graph?
Find the slope of the position-time graph at the point of interest, by drawing a tangent.
Why is the average velocity not enough?
It hides how the velocity changed during the interval, which can matter for safety or accuracy.
How can I find instantaneous velocity from a graph?
Draw the tangent to the position-time curve at that instant. Its slope is the instantaneous velocity.
Related topics in Kinematics
Distance vs displacement · Speed vs velocity · Average velocity · Acceleration · Uniform motion · Position–time graphs