What is the difference between average acceleration and instantaneous acceleration?
1D Kinematics: Average vs. Instantaneous Acceleration
Average acceleration comes from two speedometer readings. Read the speedometer at t1 and again at t2 (the car only moves forward, so each reading is its velocity), and divide the change by the time between: aavg = Δv ÷ Δt. On the v–t graph that is the slope of the orange secant through the two points. It belongs to the whole interval, not to any one instant inside it.
Instantaneous acceleration is the value the average closes in on. Press SHRINK Δt: t2 slides back toward t1, the secant pivots onto the teal tangent, and aavg settles on ainst, the slope of the tangent. Δv and Δt both shrink toward zero, but their ratio does not: the close-up keeps the triangle the same size on screen while its slope settles onto the tangent’s. The interval stops at 0.01 s, because 0 ÷ 0 is not an acceleration.
A curved v–t graph means the acceleration is changing. The car’s acceleration fades as it nears 20 m/s, so the graph bends over and the average over any interval that starts at t1 is less than ainst at t1. Only on a straight v–t line does the secant lie on the tangent for every Δt: that is constant acceleration, the one case v = v0 + at and the other kinematic equations describe.
From Average to Instantaneous Acceleration
In AP Physics 1, acceleration tells us how quickly velocity is changing. But there are two kinds worth distinguishing.
Average acceleration over a time interval is simply the total change in velocity divided by the total time elapsed — Δv/Δt. On a velocity-time graph, this is the slope of the straight line (the secant) connecting two points on the curve.
Instantaneous acceleration is the acceleration at one exact moment in time. To find it, we shrink the time interval Δt smaller and smaller. As Δt → 0, the second point slides toward the first, and the secant line rotates until it becomes a tangent — touching the curve at just one point. The slope of that tangent is the instantaneous acceleration at that moment.
Mathematically, this is the derivative: a(t) = dv/dt. Even if you haven't taken calculus yet, this simulation makes the idea visual and intuitive — the limit process is something you can see happening.
How to Use This Simulation
- Choose a velocity function using the buttons at the top — each gives the curve a different shape.
- Drag the slider to the right to set a large Δt. Notice the blue dashed secant line connecting the two points, and watch the values of Δt, Δv, and average acceleration update.
- Slowly drag the slider left to shrink Δt. Watch the second point (blue dot) creep toward the first (red dot), and observe how the secant rotates.
- Hit "Animate Δt → 0" to watch the full transition automatically — the secant becomes a solid tangent line and the display switches to instantaneous acceleration.
- Move t₁ using the arrow buttons to explore instantaneous acceleration at different points on the curve.
Key insight: When Δt is large, average acceleration is just an approximation. As Δt shrinks toward zero, that approximation becomes exact — and that exact value is the instantaneous acceleration.
