Understanding the Race Problem

This problem involves calculating the time difference between two runners, A and B, in a 1000 m race under a specific condition where runner B's speed changes mid-race due to an injury. We are given initial information from a different race that helps us determine the speeds of A and B.

Initial Race Analysis (1000 m)

In the first scenario, A beats B by 100 m or 10 seconds in a 1000 m race. This tells us two important things:

• When A finishes the 1000 m race, B is 100 m behind, meaning B has run 1000 m - 100 m = 900 m.
• The time difference between A finishing and B finishing is 10 seconds. Since A beats B, B takes 10 seconds more than A to complete the 1000 m race.

From the fact that B is 100 m behind when A finishes, and this distance represents a time difference of 10 seconds, we can determine B's speed.

Calculating Speeds

B runs the distance of 100 m (the distance by which A beats B) in 10 seconds. Therefore, B's speed (\(v_B\)) can be calculated as:

\( v_B = \frac{\text{Distance}}{\text{Time}} \)

\( v_B = \frac{100 \text{ m}}{10 \text{ s}} = 10 \text{ m/s} \)

Now we can find the time B takes to complete the 1000 m race at this speed:

\( \text{Time taken by B for 1000 m} = \frac{\text{Distance}}{\text{Speed}} \)

\( \text{Time taken by B for 1000 m} = \frac{1000 \text{ m}}{10 \text{ m/s}} = 100 \text{ seconds} \)

Since A beats B by 10 seconds, A takes 10 seconds less than B to complete the 1000 m race:

\( \text{Time taken by A for 1000 m} = \text{Time taken by B for 1000 m} - 10 \text{ seconds} \)

\( \text{Time taken by A for 1000 m} = 100 \text{ s} - 10 \text{ s} = 90 \text{ seconds} \)

We can also calculate A's speed (\(v_A\)), although it's not strictly necessary for the final answer, it's good to know:

\( v_A = \frac{1000 \text{ m}}{90 \text{ s}} = \frac{100}{9} \text{ m/s} \)

Second Race Scenario: B's Injury

Now consider the second race, also 1000 m, where B gets injured. The injury occurs after B runs a distance that is "50 m less than half the race length".

Half the race length is \( \frac{1000 \text{ m}}{2} = 500 \text{ m} \).

The distance B runs before getting injured is \( 500 \text{ m} - 50 \text{ m} = 450 \text{ m} \).

For the first 450 m, B runs at his original speed of 10 m/s.

After running 450 m, B's speed gets halved. His new speed becomes \( \frac{10 \text{ m/s}}{2} = 5 \text{ m/s} \).

The remaining distance for B to run is \( 1000 \text{ m} - 450 \text{ m} = 550 \text{ m} \).

Calculating B's Total Time in the Second Race

B's total time in the second race is the sum of the time taken for the first 450 m and the time taken for the remaining 550 m.

Time for the first 450 m (at 10 m/s):

\( \text{Time}_1 = \frac{450 \text{ m}}{10 \text{ m/s}} = 45 \text{ seconds} \)

Time for the remaining 550 m (at 5 m/s):

\( \text{Time}_2 = \frac{550 \text{ m}}{5 \text{ m/s}} = 110 \text{ seconds} \)

B's total time for the 1000 m race is:

\( \text{Total Time}_B = \text{Time}_1 + \text{Time}_2 = 45 \text{ s} + 110 \text{ s} = 155 \text{ seconds} \)

Calculating A's Time in the Second Race

A runs the 1000 m race at his constant speed (\(v_A = \frac{100}{9} \text{ m/s}\)). His time for the 1000 m race remains the same as calculated from the first scenario.

\( \text{Total Time}_A = 90 \text{ seconds} \)

Finding the Time Difference

To find by how much time A beats B, we subtract A's total time from B's total time:

\( \text{Time Difference} = \text{Total Time}_B - \text{Total Time}_A \)

\( \text{Time Difference} = 155 \text{ s} - 90 \text{ s} = 65 \text{ seconds} \)

Therefore, A will beat B by 65 seconds in the second race.

Revision Table: Key Calculations

Additional Information on Race Problems

Race problems often test your understanding of speed, time, and distance relationships. Key concepts include:

• Speed, Distance, Time Formula: \( \text{Speed} = \frac{\text{Distance}}{\text{Time}} \). This can be rearranged to find distance (\( \text{Distance} = \text{Speed} \times \text{Time} \)) or time (\( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \)).
• Head Start: A head start can be given in terms of distance (e.g., A gives B a 10 m head start) or time (e.g., A gives B a 5 second head start). This affects the distance or time each person needs to cover relative to the other.
• Beating by Distance/Time: If A beats B by 'x' meters or 't' seconds in a race of 'D' meters:
  • When A covers D meters, B covers (D-x) meters.
  • A takes 'T' time to finish, B takes (T+t) time to finish.
  • The distance 'x' is covered by B in time 't'. This can be used to find B's speed (\( v_B = x/t \)).
• Relative Speed: Sometimes, thinking about the difference in speeds can simplify problems, especially when dealing with varying distances or times.

Complex problems, like this one, involve combining these concepts and sometimes breaking the race into different segments with different speeds.