In a 2000 m race, vehicle A gives vehicle B a start of 300 m and still wins…

Concept

In a race with a "start", the runner receiving the start needs to cover only (race distance − start) while the other runs the full distance in the same time window. If the race ends in a dead heat, both take equal time to finish their respective distances; if one wins by t seconds, the other's time exceeds the winner's time by t seconds. Equating these times for two different start values gives two equations in the two speeds, which can be solved together.

Application

Let the speed of vehicle A be a m/s and the speed of vehicle B be b m/s over the 2000 m race.

1. With a start of 500 m, B only has to cover 2000 − 500 = 1500 m while A covers the full 2000 m, and the race ends in a dead heat, so both take the same time: 2000/a = 1500/b.

2. This gives the speed ratio a/b = 2000/1500 = 4/3, i.e., b = (3/4)a.

3. With a start of 300 m, B only has to cover 2000 − 300 = 1700 m, and A wins by 10 seconds, so B's time exceeds A's time by 10 seconds: 1700/b − 2000/a = 10.

4. Substituting b = (3/4)a: 1700 / ((3/4)a) − 2000/a = 10, which simplifies to (6800/3 − 2000)/a = 10, i.e., 800/(3a) = 10.

5. Solving, 3a = 80, so a = 80/3 m/s.

6. Vehicle A's time to run 2000 m = 2000 ÷ (80/3) = 2000 × 3/80 = 75 seconds = 1 minute 15 seconds.

Cross-check

With a = 80/3 m/s, b = (3/4)(80/3) = 20 m/s. In the 300 m-start race, B covers 1700 m in 1700/20 = 85 s while A covers 2000 m in 75 s — a winning margin of 85 − 75 = 10 seconds, matching the given condition. In the 500 m-start race, B covers 1500 m in 1500/20 = 75 s, exactly equal to A's 75 s, confirming the dead heat. Both conditions check out, so A takes 1 minute 15 seconds to run 2000 m.