Unwrap the Loop: Meeting and Catch-Up
环形路上的相遇与追及
Turn a closed track into a straight relative-motion story. Learn when to add speeds, when to subtract them, and how to identify the exact distance that must be closed.
Read the track before doing any arithmetic
Throughout this lesson, travelers move at constant speeds without stopping or turning around. They start moving simultaneously; when a question says they start together, they also share a starting point. Count the first meeting or catch after time 0 unless another event is specified. Measure every lead and gap along the track, in the stated direction.
Cut the loop at the start line and lay it flat. One full lap becomes one segment of length C.
Same direction: catch-up
The faster traveler gains on the slower traveler.
relative speed = fast speed − slow speedOpposite directions: meeting
The travelers close the route between them together.
relative speed = speed 1 + speed 2Choose the correct language
Same direction: the faster traveler gains one lap
When two travelers leave the same point in the same direction, the starting instant is not counted as a catch. The faster traveler must gain one full circumference before the next catch.
Build the catch-up equation
Opposite directions: their distances add
From the same start point, the travelers meet again after their combined distances make one full circumference.
Build the meeting equation
Use one race to reveal the track, then solve a new race
Two runners start together from the same point at 250 m/min and 200 m/min in the same direction. Their first catch after starting occurs at 45 minutes. Then they start together in opposite directions.
The faster runner gains one lap in 45 minutes.
Now their combined distance reaches that same circumference.
Complete the two-stage reasoning
A faster traveler who starts ahead must gain the rest of the lap
On a 400 m track, the slower runner travels 80 m/min. The faster runner travels 1.25 times as fast and starts 100 m ahead along their common direction of travel.
The 100 m lead is already behind the faster runner. To catch from behind, the faster runner must gain the remaining 300 m.
Solve the guided practice exactly
Use the stated 100 m lead, regardless of the lead slider.
Choose the exact distance that must close
The speed rule is only half the problem. The other half is choosing the correct route distance.
Find the next catch after running begins on a 500 m track.
They move in the same direction.
They move on a 500 m track.
The approaching arc is 120 m.
Enter the relative distance for each case
Relative-motion laboratory
Use the same numbers in different directions and watch the operation change.
Scenario explorer
Record the three scenario results
Error detective
Strong solvers do not only know a formula. They know why a tempting formula is wrong.
Repair four common mistakes
Meeting-and-catch-up workshop
Solve at least six of the eight problems correctly. Units are shown beside each answer field.
1 · Opposite directions
A 420 m track. Speeds 60 and 45 m/min. Same start. Meeting time?
minutes2 · Same direction
A 540 m track. Speeds 75 and 45 m/min. Same start. Next catch time?
minutes3 · Guided practice
400 m track; 100 m/min runner starts 100 m ahead of an 80 m/min runner. Catch time?
minutes4 · Recover the track
Runners start together from the same point at 250 and 200 m/min in the same direction. Their first catch after starting is at 45 min. Circumference?
metres5 · Reuse that track
Using the previous track, they start together in opposite directions. Meeting time?
minutes6 · Consecutive meetings
On a 400 m track, opposite-direction meetings are 40 s apart. A runs 6 m/s. B's speed?
m/s7 · Where is the catch?
On a 360 m track, speeds are 305 and 275 m/min from the same start and direction. At the first catch after starting, what is the distance from the start to the catch point, measured along their direction of travel and less than one lap?
metres along the track8 · Find both speeds
On a 500 m track, two runners start together from the same point in each of two trials, keeping their speeds unchanged. Their first meeting in opposite directions is after 1 min; their first catch in the same direction is after 10 min. Find the faster speed.
m/minExit ticket
Try all five questions, choosing the route gap and relative speed before calculating.
Certificate of mathematical reasoning
Loop-Unwrapping Navigator
This certifies that the learner can distinguish meeting from catch-up, select the correct route gap, and use relative speed on a closed track.
Lesson 7.1 · Grade 5 Enrichment
Teaching notes
The central two-stage example uses the chapter's opening problem: 250 m/min and 200 m/min, a same-direction catch after 45 minutes, circumference 2250 m, and an opposite-direction meeting after 5 minutes. The 400 m guided practice uses the worked example values 80 m/min, a 1.25 speed factor, and a 100 m initial lead. Selected workshop problems adapt the chapter exercise set.
The exit-ticket reflection is intentionally not scored. All automatic scoring uses numerical or selectable responses only.