A and B run a 100 metre race. A runs at a constant speed of 5 km/h and gives B a head start of 8 metres, yet still finishes 8 seconds before B. What is B's speed in km/h?

Introduction / Context:
This is a relative speed problem in a race where one runner gives the other a head start and still wins. A has a known constant speed and gives B an 8 metre start in a 100 metre race. A still finishes 8 seconds before B. From this, you must determine B's speed. The question tests your ability to convert units, calculate time from speed and distance, and interpret race time differences correctly.

Given Data / Assumptions:

• Race distance for A = 100 m.
• B gets a start of 8 m, so B runs only 92 m.
• A's speed = 5 km/h.
• A finishes 8 seconds earlier than B.
• Speeds are constant for both runners.
• We need B's speed in km/h.

Concept / Approach:
We use:

• Convert A's speed from km/h to m/s.
• Compute A's time to run 100 m.
• Since A finishes 8 seconds before B, B's time is A's time plus 8 seconds.
• B's speed = distance / time = 92 m / B's time in seconds.
• Convert B's speed back to km/h.

Careful unit conversion is critical for accuracy.

Step-by-Step Solution:
Step 1: Convert A's speed from km/h to m/s. Step 2: 5 km/h = 5 * 1000 / 3600 m/s = 5000 / 3600 m/s ≈ 1.3889 m/s. Step 3: Time taken by A to run 100 m: T_A = distance / speed = 100 / (5000 / 3600) = 100 * 3600 / 5000 = 72 seconds. Step 4: A finishes 8 seconds earlier than B, so B's time T_B = T_A + 8 = 72 + 8 = 80 seconds. Step 5: B's distance = 92 m (because of 8 m head start). Step 6: B's speed in m/s: v_B = 92 / 80 = 1.15 m/s. Step 7: Convert v_B to km/h: v_B_kmh = 1.15 * 3600 / 1000 = 1.15 * 3.6 = 4.14 km/h. Step 8: Therefore, B's speed is approximately 4.14 km/h.

Verification / Alternative check:
Check times with computed speeds:

• A's speed ≈ 1.3889 m/s, so time for 100 m is 100 / 1.3889 ≈ 72 s (as computed).
• B's speed ≈ 1.15 m/s, so time for 92 m is 92 / 1.15 ≈ 80 s.
• Difference in finishing times = 80 - 72 = 8 seconds, which matches the given information.

This confirms that our conversion and calculations are consistent with the conditions of the race.

Why Other Options Are Wrong:
5.15 kmph is greater than A's speed, which would imply B could not finish after A if given a head start.
4.25 kmph and 4.4 kmph do not yield the exact 8 second difference when you compute the times. They produce finishing time gaps different from 8 seconds and therefore do not satisfy the race conditions exactly.
Only about 4.14 kmph results in B finishing exactly 8 seconds after A when B runs 92 m and A runs 100 m.

Common Pitfalls:
Common errors include forgetting to convert km/h to m/s correctly, misinterpreting the head start so that both runners are assumed to run 100 m, or incorrectly adding the 8 seconds. Some students also mistakenly assume that B's distance is 100 - 8 = 92 without linking it to B's total time carefully. Always verify that the time difference derived from speeds and distances matches the given condition.

Final Answer:
Runner B's speed in the race is approximately 4.14 km/h.