Lesson 7.2 — First, Second, and Nth Encounters
第7讲 · 环形路上的行程问题
Learn how one encounter becomes a repeating schedule. Track the first gap, add one full relative lap for every later event, and decide carefully whether the starting instant or ending instant is counted.
What counts as the first encounter?
Assume constant speeds, simultaneous starts, and no stops or reversals throughout this lesson. “Start together” means at the same point and time. Before calculating, decide what the words count. Most problems that say “the first time they meet again” do not count the starting instant.
The encounter ladder
Imagine one traveler standing still. The other moves at relative speed. Each new encounter after the first requires one more complete relative lap.
Check the counting language
Repeated same-direction catches
Two runners start together and move in the same direction on a 420 m track. Their speeds are 75 m/min and 55 m/min.
At every catch, the faster runner has gained exactly n complete laps relative to the slower runner.
Build the schedule
Repeated opposite-direction meetings
Two runners start together on a 360 m track and move in opposite directions at 50 m/min and 40 m/min.
Their distances add. After every meeting, one more complete circumference must be covered together.
Build the opposite-direction schedule
When the starting points are opposite
Now the runners begin at opposite ends of a diameter on a 240 m track and run toward one another at 30 m/min and 20 m/min.
The first gap is only half a lap
At the start, the directed arc between the runners is:
240 ÷ 2 = 120 mAfter the first meeting, every later meeting adds one full 240 m relative lap.
Use the odd half-lap pattern
Use consecutive meetings to recover a speed
On a 400 m track, two runners move in opposite directions. The time from one meeting to the next is 40 seconds. Runner A moves at 6 m/s.
The first meeting location does not matter. From any meeting to the next meeting, the combined relative distance is exactly one circumference.
Work backward
Recover the missing speed
Count encounters inside a time window
Suppose runners start at the same point at time 0, with later same-direction catches at 15, 30, 45, 60, … minutes. Counting depends on whether the starting instant and the ending instant are included.
List the actual event times before counting. This prevents off-by-one mistakes.
Count carefully
Use each time window written below, not the current counter controls.
Meeting anywhere is not the same as meeting at the start
On a 360 m track, A runs at 60 m/min clockwise and B runs at 45 m/min counterclockwise. They begin together at the start.
Use each runner's lap time
LCM means least common multiple: A returns after 6, 12, 18, 24, … minutes; B after 8, 16, 24, … . Their first common positive return time is 24. They meet somewhere on the track every:
360 ÷ (60 + 45) = 24/7 minutesSo their meeting at the start after 24 minutes is the 7th meeting after they begin.
Distinguish the two questions
Recover two speeds from two encounter times
Two runners make two trials on the same 400 m track at unchanged speeds. In each trial they start together from the same point. Their first meeting after starting in opposite directions takes 1 minute; their first catch after starting in the same direction takes 10 minutes.
Clue 1 gives the sum
vfast + vslow = 400 ÷ 1 = 400Clue 2 gives the difference
vfast − vslow = 400 ÷ 10 = 40Sum-and-difference bars
fast = (sum + difference) ÷ 2slow = (sum − difference) ÷ 2Recover both speeds
Nth-encounter workshop
Solve at least six of eight. Before calculating, identify the first relative gap and the repeated interval.
1 · Third catch
480 m track, same start and direction, speeds 90 and 70 m/min. Time of the third catch?
minutes2 · Fifth meeting
600 m track, same start, opposite directions, speeds 80 and 70 m/min. Time of the fifth meeting?
minutes3 · Opposite starting points
300 m track; runners start at opposite ends of a diameter and run in opposite directions, combined speed 50 m/min. Time of the second meeting?
minutes4 · Missing opposite speed
400 m track; consecutive opposite meetings are 50 s apart. One speed is 3 m/s. Find the other.
m/s5 · Missing faster speed
360 m track; same-direction catches are 12 min apart. Slower speed is 18 m/min. Find the faster speed.
m/min6 · Count encounters
Runners start together at time 0 and meet again at 8, 16, 24, … minutes. How many occur in the first 35 minutes, not counting time 0?
encounters7 · Back at the start
420 m track; runners start together at the starting point in opposite directions at 70 and 60 m/min. First positive time both return to the start?
minutes8 · Recover two speeds
On a 540 m track, two trials use unchanged speeds and start together from the same point. The first opposite-direction meeting is after 2 min; the first same-direction catch is after 18 min. Faster speed?
m/minExit ticket
Try all five questions using an encounter schedule. The optional reflection is not scored.
Certificate of mathematical reasoning
Nth-Encounter Scheduler
This certifies that the learner can build repeated meeting and catch schedules, count events correctly, and distinguish meeting anywhere from returning to a named point.
Lesson 7.2 · Grade 5 Enrichment
Teaching notes
It uses the chapter's same-direction and opposite-direction relative-speed framework, the first-versus-second meeting structure from the diameter-endpoint example, and exercise patterns involving consecutive encounter intervals, counting repeated encounters, returning to a start point, and recovering two speeds from a sum and a difference.
The student page is self-contained.All automatically graded responses are numbers or selections; the reflection is intentionally ungraded.