Race Distance Calculation: P vs Q Speed Ratio

This solution details the calculation for the distance between runners P and Q when P finishes a 500 m race, considering their speed ratio and a staggered start for Q.

Step 1: Define Speeds Based on Ratio

Let the speeds of runner P and runner Q be $3x$ m/s and $4x$ m/s, respectively, based on the given ratio of 3:4.

Step 2: Calculate Time for P to Cover Initial Distance

P has already covered 140 m when Q starts. The time taken by P to cover this initial distance is:

$t_{P_{140}} = \frac{\text{Distance}}{\text{Speed}_P} = \frac{140 \text{ m}}{3x \text{ m/s}} = \frac{140}{3x} \text{ seconds}$

Step 3: Determine Time P Takes to Finish the Race

The total race distance is 500 m. The total time P takes to complete the race from the start is:

$T_P = \frac{\text{Total Distance}}{\text{Speed}_P} = \frac{500 \text{ m}}{3x \text{ m/s}} = \frac{500}{3x} \text{ seconds}$

Step 4: Calculate Duration Q Runs Until P Finishes

Q starts at time $t_{P_{140}}$. P finishes at time $T_P$. The duration Q runs is the difference between these times:

$ \text{Duration}_Q = T_P - t_{P_{140}} = \frac{500}{3x} - \frac{140}{3x} = \frac{360}{3x} = \frac{120}{x} \text{ seconds} $

Step 5: Calculate Distance Covered by Q

In the duration calculated above, Q covers a distance using their speed:

$ \text{Distance}_Q = \text{Speed}_Q \times \text{Duration}_Q = (4x \text{ m/s}) \times \left(\frac{120}{x} \text{ seconds}\right) = 480 \text{ m} $

Step 6: Find the Distance Between P and Q at P's Finish

When P finishes the race, P is at the 500 m mark. Q has covered 480 m. The distance between them is:

$ \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.