Find Time for Alice to Lap Bob on Circular Track

What You Will Learn

• How to calculate circumference when radius contains π
• Finding relative speed for objects moving in the same direction
• Understanding what "lapping" means in circular motion problems
• Converting between different measures of circular motion
• Setting up and solving multi-step motion problems

Solution: Method 1 — The Relative Speed Approach

This problem is all about understanding that "lapping" means gaining a full circumference worth of distance. Let's work through it systematically.

Step 1 — Calculate the track circumference

Using the formula C = 2πr with radius 100/π meters:

The π terms cancel out perfectly, giving us a clean circumference of 200 meters.

Step 2 — Find the relative speed

Since both runners move in the same direction, Alice gains ground on Bob at the difference of their speeds:

This means Alice gains 4 meters on Bob every second.

Step 3 — Calculate time for one lap

For Alice to lap Bob once, she must gain exactly one full circumference (200 m) on him:

Step 4 — Find time for three laps

Since lapping events happen at regular intervals:

Solution: Method 2 — The Position Tracking Method

Instead of thinking about relative speed, we can track each runner's position and find when Alice is exactly one, two, and three laps ahead.

Step 1 — Express positions as functions of time

After t seconds:

Step 2 — Set up the lapping condition

Alice laps Bob when she has traveled exactly one more complete track length than Bob:

where n is the number of complete laps Alice is ahead.

Step 3 — Solve for each lap

Rearranging the equation:

Step 4 — Find the specific times

Verification

Let's check our answer by calculating how far each runner has traveled after 150 seconds:

Since the track is 200 meters around, a 600-meter difference means Alice is exactly 3 complete laps ahead of Bob. ✓

We can also verify the individual lapping times:

Common Pitfalls

The General Formula

For any circular track lapping problem where both objects move in the same direction:

where v₁ is the faster speed and v₂ is the slower speed.

This formula works because:

• The relative speed (v₁ - v₂) tells us how quickly the faster object gains ground
• Each lap requires gaining exactly one circumference worth of distance
• The lapping events occur at regular intervals

How to Spot This Problem Type

Watch for these key phrases that signal a relative motion problem:

• "laps" or "overtakes" — indicates one object gaining a full circuit on another
• "circular track" or "around a loop" — tells you it's circular motion
• "same direction" vs "opposite directions" — determines whether to add or subtract speeds
• "how long until..." — asking for time, not distance

This problem type also appears in disguised forms:

• Cars on a racetrack
• Cyclists on a velodrome
• Satellites orbiting at different speeds
• Clock hands (hour vs minute hand)