Switching Routes and Connected Loop Networks
变轨与相连环路
A switch may change before a traveler reaches it, and two travelers may share only part of a network. Learn to track the current route, the selected route, the next junction, and every encounter.
A route selector is not the same as the route being traveled
On a network, the switch can select a new loop while the train is still far from the junction. The train changes routes only when it actually reaches the junction.
Four pieces of state
At any moment, record:
The selector tells us which route is ready. The actual route tells us where the train is moving now.
The junction rule
A track switch cannot pull a moving train sideways from the middle of a loop.
So every route-change problem is an event problem, not just a speed problem.
The dashed route is selected. The solid route is the one currently being traveled.
Check the rule
Convert loop length into lap time
For Missions 2–5, the train moves continuously at a constant 10 metres per minute, without waiting at the junction. One loop is 3 metres long and the other is 1.5 metres long.
original values
Converting to seconds makes the one-minute switch schedule much easier to track.
Calculate the two lap times
Use lengths 3 m and 1.5 m and speed 10 m/min, regardless of the machine controls. Later train questions use these same fixed values.
Follow the train through the first route change
The train leaves junction A at time 0 on the selected large loop. The selector alternates every 60 seconds: small at 60 s, large at 120 s. At each arrival at A, the train immediately follows the selected loop. Time 0 is not a return. At 60 s the train is still between junction visits.
Move the slider across the selector change at 60 seconds and the junction arrival at 72 seconds.
Build the first event
Count the tenth return to A
The train enters the small loop at 72 seconds. It returns every 9 seconds. At 120 seconds the selector changes again, but the train must finish its current small lap.
Returns on the large loop
The train returns at:
The return at 72 seconds is return number 4.
Returns on the small loop
After switching at 72 seconds, it returns at:
The return at 126 seconds is return number 10.
Complete the count
Build an event-driven route-state table
A reliable solution records what the selector says and what the train is actually doing at each important time.
Read the state, not just the timer
Map two figure-eight routes on three connected loops
Each circle has circumference 210 cm. Caterpillar A moves at a constant 20 cm/min and B at 15 cm/min, starting simultaneously at the marked outermost points. A first follows the lower half of loop 1 to C, then one full clockwise lap of loop 2. B first follows the lower half of loop 3 to D, then one full counterclockwise lap of loop 2. Each then returns to its own outer loop and continues the figure-eight route without stopping. Only loop 2 and its junctions are shared.
A starts at the far-left point A. B starts at the far-right point B. They can meet only while both occupy the shared middle loop or one of its junctions.
A’s route
A reaches junction C after half of loop 1:
B’s route
B reaches junction D after half of loop 3:
Build the route map
Find the first meeting on the shared loop
At 7 minutes, B has just reached D. A has already been on the middle loop for 1.75 minutes.
The lower number line unwraps the 210-cm middle circle from C back to C.
Calculate the first meeting
Find the second meeting before either traveler leaves the shared loop
After the first meeting, the next meeting needs one full relative lap of the shared 210-cm circle.
A stays on loop 2 until 15.75 minutes. B stays there until 21 minutes, so the second meeting at 15 minutes is valid.
Time between meetings
First meeting at 9 minutes:
Total distance since time 0
By the second meeting, caterpillar A has crawled 300 centimetres.
Complete the second meeting
Switching-route and loop-network workshop
Solve at least six of the eight questions. These combine lap times, delayed route changes, shared-loop entry, and repeated meetings.
1. A 4-m loop is traveled at 12 m/min. One lap takes how many seconds?
2. A 2-m loop at the same speed takes how many seconds?
3. A train is on the 20-second loop. The selector changes at 45 seconds. The train last passed the junction at 40 seconds. At what time can it actually switch?
4. In the fixed train scenario from Missions 2–5, which route is the train actually traveling at 60 seconds? Enter large or small.
5. At what second does the train in Missions 2–5 return to A for the tenth time?
6. Two circles of circumference 210 cm make one figure-eight route. What is one full route length?
7. In the three-loop problem, at what minute do the caterpillars first meet?
8. How far has caterpillar A crawled by the second meeting?
Exit ticket
Check the five answers, then compare any uncertain steps in worked review. The optional reflection is not automatically graded.
Certificate of mastery
Route-Network State Navigator
This certifies that the learner can track route selectors, actual routes, junction events, figure-eight paths, and repeated meetings on connected loop networks.
Teaching notes
The train investigation is based on Chapter 7, Example 4, and the three-loop caterpillar investigation is based on Guided Practice 4.
The worked example train returns to A for the tenth time after 126 seconds, or 2.1 minutes. In the three-loop problem, the caterpillars meet at 9 minutes and 15 minutes; by the second meeting, caterpillar A has traveled 300 cm.