Average Velocity — Definition, Formulae, Practice Problems & FAQs
Guide to average velocity: definition, formulas for common cases (equal displacements, equal times, mixed cases), solved examples, and frequently asked questions.
Velocity — quick reminder
Velocity \(v\) is a vector that describes how fast and in what direction an object’s position changes. For straight-line motion: \[ v = \frac{\text{displacement}}{\text{time taken}}. \] SI unit: metres per second (m/s). Note: displacement is a vector (it cares about direction), while distance is a scalar.
Average velocity — definition
The average velocity over a time interval is defined as the total displacement divided by the total time: \[ v_{\text{avg}} \;=\; \frac{\text{total displacement}}{\text{total time}}. \]
- Average speed = (total distance) / (total time) — scalar.
- Average velocity = (total displacement) / (total time) — vector (may be zero if displacement is zero).
Average velocity — common formulae
Below are useful formulas for different scenarios.
1) Motion in successive displacements \(s_1,s_2,\dots\) with speeds \(v_1,v_2,\dots\)
Total displacement \(S = s_1 + s_2 + \dots\). Time for each segment \(t_i = s_i/v_i\). Therefore \[ v_{\text{avg}} = \frac{s_1+s_2+\dots}{\dfrac{s_1}{v_1}+\dfrac{s_2}{v_2}+\dots}. \]
2) Equal displacements (each segment length = \(s\)) with speeds \(v_1\) and \(v_2\)
For two equal-distance segments: \[ v_{\text{avg}} = \frac{2v_1 v_2}{v_1 + v_2} \] (the harmonic mean of \(v_1\) and \(v_2\)). This is the formula you use when the trip is split into equal displacements at different speeds.
3) Equal time intervals \(t\) with speeds \(v_1,v_2,\dots,v_n\)
If the object travels with speed \(v_1\) for time \(t\), then \(v_2\) for time \(t\), etc., total displacement \(= t(v_1+v_2+\dots+v_n)\) and total time \(= nt\). Hence \[ v_{\text{avg}} = \frac{v_1 + v_2 + \dots + v_n}{n} \] (the arithmetic mean of the speeds).
4) Mixed example (half displacement at \(v_0\); remaining half of displacement covered in two equal time intervals at \(v_1\) and \(v_2\))
Let total displacement be \(S\). First half: \(S/2\) at speed \(v_0\) takes time \(t_1 = \dfrac{S}{2v_0}\). The remaining half \(S/2\) is covered in total time \(T\), where the first half of that time uses speed \(v_1\) and the second half uses \(v_2\). The distance covered during the two equal sub-times is \((T/2)(v_1+v_2)\) which must equal \(S/2\). Thus \[ T = \frac{S}{v_1 + v_2}. \] Total time = \(t_1 + T = \dfrac{S}{2v_0} + \dfrac{S}{v_1+v_2}\). Therefore \[ v_{\text{avg}} = \frac{S}{\dfrac{S}{2v_0} + \dfrac{S}{v_1+v_2}} \;=\; \frac{2v_0 (v_1+v_2)}{\,v_1+v_2 + 2v_0\,}. \]
Practice problems — worked solutions
Important distinction: if he starts and finishes at the same point (home), his total displacement is zero, so the average velocity for the complete trip is
If the question instead asks for average speed (total distance / total time) — which many exam-style versions do — calculate:
Answer: average velocity = 0 (vector); average speed ≈ 26.67 km/h.
Using the derivation in the formula section, the average velocity is
Answer: \(\displaystyle v_{\text{avg}}=\dfrac{2v_0 (v_1+v_2)}{v_1+v_2 + 2v_0}.\)
Use harmonic-mean formula for equal distances:
Answer: 48 km/h.
Displacement from A to B is a straight line equal to the diameter: \(|\Delta \mathbf{r}| = 2R = 40\ \text{m}.\) The path length along a semicircle is \(\pi R\). Time taken \(t = \dfrac{\text{arc length}}{v}=\dfrac{\pi R}{v} = \dfrac{\pi\times20}{10} = 2\pi\ \text{s}\) (≈ 6.2832 s). Hence average velocity magnitude:
Answer: \(v_{\text{avg}}=\dfrac{20}{\pi}\ \text{m/s}\approx 6.37\ \text{m/s}.\)
FAQs
Q: What is the difference between average speed and average velocity?
A: Average speed = total distance / total time (scalar). Average velocity = total displacement / total time (vector). If the start and end point are the same, average velocity = 0 while average speed is not necessarily zero.
Q: Can average velocity be zero even if an object was moving?
A: Yes — if total displacement over the interval is zero (for example, a round trip), average velocity is zero despite nonzero motion during the trip.
Q: When do we use the harmonic mean vs arithmetic mean?
A: Use the harmonic mean (e.g., \(2v_1v_2/(v_1+v_2)\)) when the object covers equal distances at different speeds. Use the arithmetic mean when the object spends equal times at each speed.
Q: Why is average velocity sometimes negative?
A: Average velocity carries direction. If total displacement is in the negative coordinate direction, the average velocity will be negative even if instantaneous velocities were positive at some times.
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