7.2 Relative Speed, Catch-Up & Current/Wind Problems

Add Speeds or Subtract Them

Relative-motion stems on GMAT Focus Quant still live in OG Math Review 3.3: a rate is a ratio, and relative rate is the rate at which a gap shrinks or grows. Instead of writing two position functions of t, freeze one reference frame and treat the pair as a single uniform-motion problem D = R_rel × T.

Two geometric facts decide whether you add or subtract:

• Opposite directions (toward each other or away from each other): the gap changes at the sum of the speeds. R_rel = R₁ + R₂.
• Same direction (chase or overtake): the gap changes at the difference. R_rel = R_fast − R_slow.

Time is then T = (relevant distance) / R_rel. The relevant distance is the initial separation for a meeting, the target separation for a diverging pair, or the head-start lead for a catch-up.

Worked example (converging, simultaneous start). Towns 240 miles apart. One car leaves each town at the same moment, 40 mph and 80 mph, toward each other. Relative closing speed 40 + 80 = 120 mph. Time to meet = 240/120 = 2 hours. In those 2 hours the 40-mph car covers 80 miles and the 80-mph car covers 160 miles; 80 + 160 = 240 confirms the meeting point.

A same-direction version of those speeds is a different problem. If both cars leave the same town on the same road, the 80-mph car pulls away from the 40-mph car at 40 mph. After 2 hours they are 80 miles apart, not together.

Catch-Up and the Freeze-the-Clock Step

Most chase stems are staggered: the slower traveler starts first. Do not use the original clock as t = 0 for both. Freeze time at the instant the faster traveler starts, compute the lead that already exists, and only then apply relative speed.

1. Head-start time Δt (convert minutes to hours).
2. Head-start distance D_lead = R_slow × Δt.
3. Relative closing speed R_rel = R_fast − R_slow.
4. Catch-up time after the second start: T = D_lead / R_rel.
5. Clock time = second start + T. Distance from the origin = R_fast × T (or R_slow × (Δt + T)); both must match.

Worked example. Car A leaves Town X at 1:00 PM at 40 mph. Car B leaves Town X at 2:30 PM at 60 mph on the same road. Head-start time is 1.5 hours, so D_lead = 40 × 1.5 = 60 miles. Relative speed 60 − 40 = 20 mph. Catch-up time 60/20 = 3 hours after 2:30 PM, so they meet at 5:30 PM. Check: by 5:30 PM, A has been moving 4.5 hours × 40 = 180 miles; B has been moving 3 hours × 60 = 180 miles.

The same protocol handles two trains on a single track when one has a station head start, a police car chasing a speeder who already has a mile of lead, and a faster runner starting a lap behind. Always convert the lead into a distance before you divide by the speed difference. A stem that says "a 10-minute head start" is not yet a distance.

If the two objects start from different points and travel the same direction, the initial gap is the along-road separation at t = 0, not a time lead. A car 50 miles ahead at 45 mph, chased by a car at 55 mph, is caught in 50/(55 − 45) = 5 hours. No freeze-the-clock step is needed because both are already moving.

Current and Wind

A boat in still water (or a plane in still air) has speed B. A current or wind of speed C adds when you travel with the medium and subtracts when you travel against it:

• Downstream / tailwind: R_down = B + C
• Upstream / headwind: R_up = B − C

Given those two effective speeds, recover the pair instantly:

• B = (R_down + R_up) / 2 (arithmetic mean — here the two rates already are speeds, not an average-speed trap)
• C = (R_down − R_up) / 2

Adding the two equations cancels C; subtracting them cancels B. Units still have to match. If upstream time is given in minutes, convert before you compute R_up = D/T.

Worked example. A motorboat travels 36 miles upstream in 3 hours and returns the same 36 miles downstream in 2 hours. Upstream rate 36/3 = 12 mph; downstream rate 36/2 = 18 mph. Still-water speed B = (18 + 12)/2 = 15 mph. Current C = (18 − 12)/2 = 3 mph. Check: 15 − 3 = 12 and 15 + 3 = 18.

A round trip of D miles upstream and D miles downstream in a current always takes longer than the same 2D miles in still water. Upstream time D/(B − C) plus downstream time D/(B + C) equals 2BD/(B² − C²), which is greater than 2D/B whenever C > 0. The boat spends more extra time fighting the current than it saves riding the current — the same weighting that made the harmonic mean slower than the arithmetic mean in Section 7.1. Do not average the upstream and downstream times to claim the still-water time; average the upstream and downstream speeds to recover B, then compute 2D/B if the question asked for still-water time.

Wind problems are identical with different nouns. A plane with airspeed 200 mph and a 40-mph wind flies 240 mph with the wind and 160 mph against it. A 480-mile out-and-back with that wind takes 480/240 + 480/160 = 2 + 3 = 5 hours, versus 480×2/200 = 4.8 hours in still air.

Circular Tracks

On a closed loop of length L the "gap" that must close for a meeting is one full lap (or n laps for the n-th meeting).

• Opposite directions: they close L at R₁ + R₂. First meeting at T = L/(R₁ + R₂). The n-th meeting at n L/(R₁ + R₂).
• Same direction: the faster runner must gain a full lap on the slower one. Time between laps T = L / |R₁ − R₂|.

Worked example. A 400-meter track. Karen at 6 m/s and Liam at 4 m/s, opposite directions, same start. Relative speed 10 m/s. First meeting in 400/10 = 40 seconds. Third meeting in 3 × 40 = 120 seconds. Combined distance in 120 seconds: 120×6 + 120×4 = 1,200 meters = exactly three laps of 400 meters.

Same runners, same direction: relative speed 2 m/s. Karen laps Liam every 400/2 = 200 seconds. Opposite-direction meetings are more frequent than same-direction laps whenever both speeds are positive — a quick reasonableness check if an answer choice for opposite directions is larger than L / |R₁ − R₂|.

When runners start at different points on the track, the first-meeting distance is the arc between them along the direction they close, not necessarily a full lap. Convert that arc to a length, then divide by relative speed. Subsequent meetings return to the full-lap interval.

Decision sheet

• Two objects, opposite headings → add.
• Two objects, same heading → subtract.
• One object delayed → freeze, then subtract (or add, if they still head toward each other).
• Medium moving → B ± C, then (sum)/2 and (difference)/2.
• Closed loop → replace the gap with L (or nL).