Race Problem Analysis: P and Q's Performance

This problem involves understanding relative speeds and distances in a race scenario. We are given the total race distance, the speed ratio of two runners, P and Q, and a head start condition. The goal is to find the distance between P and Q when P wins the race.

Understanding the Race Setup

• The total length of the race is 500 m.
• The speeds of P and Q are in the ratio of 3 ∶ 4. This means if P's speed is \(3v\), then Q's speed is \(4v\) for some unit speed \(v\).
• Q starts the race when P has already covered 140 m. This is a head start for P.
• When P wins the race, P will have covered the full 500 m distance.

Calculating P's Winning Time

When Q starts, P has already covered 140 m. This means P is 140 m ahead of the starting line, and Q is at the starting line (0 m). For P to win the race, P must cover the remaining distance to reach the 500 m mark.

• P's initial distance covered = 140 m.
• Total race distance = 500 m.
• Distance P needs to cover from the point Q starts = \(500 \text{ m} - 140 \text{ m} = 360 \text{ m}\).

Let's assume P's speed is \(3x\) units/time and Q's speed is \(4x\) units/time. To find the time taken for P to win the race from the 140 m mark, we use the formula: \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \).

Time taken by P to cover 360 m:

$$ \text{Time}_{\text{P wins}} = \frac{360 \text{ m}}{3x \text{ m/unit time}} = \frac{120}{x} \text{ unit time} $$

Determining Q's Progress

Q starts running at the same moment P is at the 140 m mark. Both P and Q run for the same duration until P crosses the finish line. We need to calculate how much distance Q covers in the time P takes to win the race.

• Q's speed = \(4x\) units/time.
• Time Q runs = \( \frac{120}{x} \) unit time (same as P's winning time).

Distance covered by Q in this time:

$$ \text{Distance}_{\text{Q covers}} = \text{Speed}_{\text{Q}} \times \text{Time}_{\text{P wins}} $$

$$ \text{Distance}_{\text{Q covers}} = 4x \text{ m/unit time} \times \frac{120}{x} \text{ unit time} $$

$$ \text{Distance}_{\text{Q covers}} = 4 \times 120 = 480 \text{ m} $$

When P wins the race (P is at 500 m), Q has covered 480 m from the starting line.

Final Distance Calculation

Now we need to find the distance between P and Q at the exact moment P wins the race.

• P's position when P wins = 500 m (the finish line).
• Q's position when P wins = 480 m (distance covered by Q from the start).

The distance between P and Q is the difference between their positions:

$$ \text{Distance between P and Q} = \text{P's position} - \text{Q's position} $$

$$ \text{Distance between P and Q} = 500 \text{ m} - 480 \text{ m} = 20 \text{ m} $$

Therefore, the distance between P and Q when P wins the race is 20 m.