Same direction
Gap closes at fast speed - slow speed.
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GMAT Quant
Relative Speed
Solve catch-up, meeting, and pass-then-separate motion problems by tracking how the gap changes.
Core equation
All motion starts with Distance = Rate x Time.
Relative speed becomes useful when the question is really about how fast the distance between two objects changes.
Direction determines whether rates subtract or add.
Opposite directions
Gap closes at speed 1 + speed 2.
Catch-up setup
A head start must become a distance gap first.
If a slower car travels for h hours before the faster car starts, head-start gap = slow speed x h.
Example: 35 mph gets a 2-hour head start; 55 mph follows.
- Initial gap
- 35*2 = 70 miles
- Relative speed
- 55 - 35 = 20 mph
- Catch time
- 70/20 = 3.5 hours
Opposite directions: the closing speeds add.
- Initial gap
- 210 miles
- Speeds
- 60 mph and 45 mph
- Relative speed
- 105 mph
- Meeting time
- 210/105 = 2 hours
Pass then separate
After two travelers pass, the gap starts growing again.
If the initial gap is G and they later are L apart after passing, total gap change is G + L. Divide by the sum of their speeds.
Recognition cues
- 01Catch up, overtake, behind, head start -> subtract speeds.
- 02Toward each other, opposite ends, meet -> add speeds.
- 03After passing, later are X apart -> initial gap + later gap.
Mini application
Try this: two runners are 90 m apart and run toward each other at 4 m/s and 5 m/s.
How long until they meet?
Solution
They meet in 10 seconds.
The gap closes at 4 + 5 = 9 m/s, so time = 90/9 = 10.
Remember this
Track the gap, not two separate journeys.
Same direction -> difference. Opposite directions -> sum.