Lesson 7 of 10
How to Find Average Velocity of a Multi-Part Trip
多段运动的平均速度
A car covers 1/4 of the trip at 20 m/s and the rest at speed v. The whole-trip average velocity is 16 m/s. What is v?
Lesson notes
A short written version of Lesson 7. The study sheet has more practice questions.
Short answer
v = 15 m/s. The trip is split by displacement, so find each part's time: t₁ = x/80 and t₂ = 3x/(4v). Then x / (t₁ + t₂) = 16 gives v = 15 m/s.
Key ideas
- Average velocity is always total displacement divided by total time, x / t. This is true for uniform or changing motion.
- In these problems the motion is one-way and in a straight line. You know the average velocity of each segment.
- Look at the whole trip: find the total displacement, find the total time, then divide.
- Split by time: use the time of each segment to find the total displacement.
- Split by displacement: use the displacement of each segment to find the total time. The math is a bit harder.
- Some problems give the whole-trip average velocity and ask for one segment. Use the same method.
Step by step
- The car goes 1/4 of the distance at 20 m/s and 3/4 at speed v. The trip is split by displacement, so find the times.
- Let the total displacement be x. The first time is t₁ = (x/4) ÷ 20 = x/80.
- The second time is t₂ = (3x/4) ÷ v = 3x/(4v).
- Average velocity = x / (t₁ + t₂). The x cancels, so it equals 80v / (v + 60).
- Set 80v / (v + 60) = 16. Multiply both sides by (v + 60): 80v = 16v + 960.
- So 64v = 960 and v = 15 m/s. The answer is C.
Common mistakes
- Wrong: averaging the segment velocities, like v/2 and 2v. Right: always divide the total displacement by the total time.
- Wrong: mixing up the two types. Right: split by time, use time to find displacement; split by displacement, use displacement to find time.
- Wrong: thinking x / t works only for uniform motion. Right: average velocity is x / t for any motion, speeding up or slowing down.
Practice
Question 1. A particle moves along a straight line, always in the same direction. In the first 2/3 of the time its average velocity is v/2; in the last 1/3 of the time it is 2v. What is the average velocity of the whole trip? ( )
A. 3v/4
B. 3v/2
C. v
D. 2v/3
Show the answer
C. Split by time. Let the total time be t. x₁ = (2t/3)(v/2) = vt/3 and x₂ = (t/3)(2v) = 2vt/3, so x = vt and average velocity = x / t = v.
Question 2. A particle moves along a straight line, always in the same direction. Its average velocity is v₁ over the first 1/3 of the displacement and v₂ over the last 2/3. What is the average velocity of the whole trip?
Show the answer
3v₁v₂ / (v₂ + 2v₁). Split by displacement. t₁ = (x/3) ÷ v₁ = x/(3v₁) and t₂ = (2x/3) ÷ v₂ = 2x/(3v₂). Average velocity = x / (t₁ + t₂); cancel x. Check: if v₁ = v₂ = v, the result is v.
Words in English and Chinese
- average velocity
- 平均速度
- total displacement
- 总位移
- total time
- 总时间
- piecewise analysis
- 分段分析
- overall analysis
- 整体分析
- one-way straight-line motion
- 单向直线运动
- split by time / by displacement
- 按时间 / 按位移分段
Sum it up before you move on.
Write this lesson in one line:
- What was the question?
- What is the simple version?
- What would you have got wrong?
Parts of the video (9)
- 0:00A car problem: find v (preview)
- 0:26Two big ideas: piecewise vs. overall analysis
- 3:38Problem features: one-way straight-line motion
- 4:49Core idea: total displacement ÷ total time
- 5:51Split by time · Example 1
- 8:17Split by displacement · Example 2
- 10:41Example 3: find one segment from the whole trip
- 13:13Recap: split by time or by displacement
- 13:50Summary