How Does Average Velocity Become Instantaneous Velocity?

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1D Kinematics: Average vs. Instantaneous Velocity

1D Kinematics: Average vs. Instantaneous Velocity

Start time t1s
Interval Δts
PHYSICS INSIGHTS

Average velocity comes from two positions. Read the tape under the car at t1 and at t2: vavg = Δx ÷ Δt, the slope of the orange secant on the x–t graph. It belongs to the whole interval, not to any one instant in it.

Instantaneous velocity is where the average settles. Press SHRINK Δt: the faded car slides back to the car at t1, the secant pivots onto the teal tangent, and vavg settles on vinst, the speedometer reading (the car only moves forward, so speed and velocity agree). Δx and Δt both shrink toward zero, but their ratio does not. The interval stops at 0.01 s, because 0 ÷ 0 is not a velocity.

Why the average runs ahead. The car speeds up the whole time, so after t1 it always moves faster than at t1: vavg stays above the speedometer reading for every Δt, and the gap shrinks with Δt.

Simulation by The Science Cube — https://www.thesciencecube.com/

What Happens to Average Velocity as the Time Interval Shrinks?

In short: As the time interval Δt shrinks toward zero, the average velocity over it settles on the instantaneous velocity: v = lim(Δt→0) Δx/Δt. On the x–t graph, the secant line between two points pivots into the tangent at t₁, and the slope of that tangent is the instantaneous velocity.

Common doubts

Why can't Δt be exactly zero? Then Δx/Δt would be 0 ÷ 0, which is not a velocity. The velocity is the value the ratio approaches as both Δx and Δt get smaller, which is why the simulation stops at 0.01 s.

Is the average velocity over an interval the same as the velocity at its start? Not when the object is speeding up. With the default settings the car speeds up, so its average velocity from t₁ to t₂ (7.00 m/s) is larger than its velocity at t₁ (4.00 m/s).

What is instantaneous velocity in everyday terms? For motion in one direction, it is what a speedometer reads at that moment. The sign gives the direction.

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