Instantaneous Velocity — Practice Problem Solutions

1. Average velocity is displacement divided by the total time over some interval — it describes the whole trip. Instantaneous velocity is the velocity at one single moment — what the object is doing right then, found by shrinking that time interval down toward zero.

2. Instantaneous velocity (well, instantaneous speed, since no direction is given). A speedometer reading is a snapshot at one moment, not an average taken over an entire trip's distance and time.

3. $v\approx\dfrac{5.10-4.90}{1.01-0.99}=\dfrac{0.20}{0.02}=10\text{ m/s}$.

4. $v\approx\dfrac{12.3-11.7}{3.1-2.9}=\dfrac{0.6}{0.2}=3\text{ m/s}$.

5. A sprinter starts from rest and speeds up over the 10 s — the velocity isn't constant. Right at $t=1\text{ s}$ the runner has barely begun accelerating out of the blocks, so their instantaneous velocity is well below the 8 m/s average that gets pulled up by the faster middle/end of the sprint.

6. a) Near $t=2\text{ s}$: $v\approx\dfrac{8.2-7.8}{2.05-1.95}=\dfrac{0.4}{0.1}=4\text{ m/s}$. Near $t=5\text{ s}$: $v\approx\dfrac{15.1-14.9}{5.05-4.95}=\dfrac{0.2}{0.1}=2\text{ m/s}$. b) The skater was moving faster at $t=2\text{ s}$ (4 m/s vs. 2 m/s). c) In the ideal mathematical sense, yes — narrower windows get closer to the true instantaneous value. But in practice, real position measurements carry some uncertainty, and an extremely narrow window (0.001 s) can make that measurement error dominate the tiny position difference, actually making the estimate less reliable — so "narrower is always better" doesn't hold once real-world measurement limits are considered.

7. With only whole-second resolution, any velocity you calculate is really an average velocity over a full one-second interval, not a true instantaneous velocity — you'd be measuring "how far it moved during that second," which blurs together whatever speeding up or slowing down happened within it.