01 / CHAPTER 03
Instantaneous velocity
Read the tangent of a position–time curve and make the derivative physically meaningful.
ANCHOR EQUATION
Instantaneous velocity
In symbols: v = dx/dt
The derivative is the slope of the position–time curve at one instant.
Velocity is a tangent
Velocity is the rate at which position changes, not a label for how much position has accumulated. On an x–t graph, the local tangent tells the object’s instantaneous direction and rate of travel.
Shrink the time interval
Over an interval Δt, average velocity is Δx/Δt = [x(t + Δt) − x(t)]/Δt. Let Δt shrink while the interval remains centered on t. The limiting slope is v(t) = dx/dt. For x(t) = 4t² − 2t + 1, differentiation gives v(t) = 8t − 2; the tangent gets steeper because the object is speeding up.
At t = 3.0 s, the model gives v = 8(3) − 2 = 22 m/s. Between 3.0 and 3.1 s, the average velocity is [x(3.1) − x(3.0)]/0.1 = 22.4 m/s, already close but not equal to the instantaneous value at the left endpoint.
Corners and noisy data
At a sharp corner in x(t), the left and right slopes differ, so instantaneous velocity is undefined at that exact instant. Real measurements also contain noise; fitting a smooth model or using a symmetric finite difference is often more reliable than dividing neighboring raw samples.
Work it: For x(t) = 2t³ − 3t², calculate the average velocity from 1.00 to 1.01 s and the exact velocity at 1.00 s. Explain the numerical difference in terms of changing velocity.