The first and the second runners in a 100 m dash have a gap of half metre at the mid way stage. By what percentage should the second runner increase his speed just to win the race, assuming that the first runner goes steady ?

Options

A. 0.01
B. 0.0102
C. 0.02
D. 0.0202

Explanation

To solve the problem step by step, we will analyze the situation of the two runners in the 100 m dash.

### Step 1: Understand the positions of the runners at the midway point
At the midway point (50 m), the first runner (A) has covered 50 m, while the second runner (B) has covered 49.5 m. This means that B is 0.5 m behind A.

### Step 2: Determine the time taken by both runners to reach the midway point
Let the speed of runner A be \( v_1 \) and the speed of runner B be \( v_2 \). The time taken by A to reach the midway point can be expressed as:
\[ t = \frac{50}{v_1} \]
For runner B, the time taken to cover 49.5 m is:
\[ t = \frac{49.5}{v_2} \]
Since both runners reach the midway point at the same time, we can equate the two expressions:
\[ \frac{50}{v_1} = \frac{49.5}{v_2} \]

### Step 3: Determine the time taken to finish the race
After reaching the midway point, runner A still has to cover another 50 m to finish the race. The time taken by A to cover this distance is:
\[ t_A = \frac{50}{v_1} \]
Runner B, who is now 0.5 m behind, needs to cover 50.5 m to finish the race. The time taken by B to cover this distance at a new speed \( v_3 \) is:
\[ t_B = \frac{50.5}{v_3} \]

### Step 4: Set the finishing times equal
For runner B to win, he must finish just before A, which means we can set the finishing times equal:
\[ \frac{50}{v_1} = \frac{50.5}{v_3} \]

### Step 5: Relate the speeds of the runners
From the two equations we have:
1. \( \frac{50}{v_1} = \frac{49.5}{v_2} \)
2. \( \frac{50}{v_1} = \frac{50.5}{v_3} \)

From the first equation, we can express \( v_1 \) in terms of \( v_2 \):
\[ v_1 = \frac{50 \cdot v_2}{49.5} \]

Substituting this into the second equation gives:
\[ \frac{50}{\frac{50 \cdot v_2}{49.5}} = \frac{50.5}{v_3} \]
Simplifying this leads to:
\[ \frac{49.5}{v_2} = \frac{50.5}{v_3} \]

### Step 6: Find the new speed \( v_3 \)
Rearranging gives:
\[ v_3 = v_2 \cdot \frac{50.5}{49.5} \]

### Step 7: Calculate the percentage increase in speed
The percentage increase in speed from \( v_2 \) to \( v_3 \) is given by:
\[ \text{Percentage Increase} = \frac{v_3 - v_2}{v_2} \times 100 \]
Substituting \( v_3 \):
\[ \text{Percentage Increase} = \frac{v_2 \cdot \frac{50.5}{49.5} - v_2}{v_2} \times 100 \]
This simplifies to:
\[ \text{Percentage Increase} = \left( \frac{50.5 - 49.5}{49.5} \right) \times 100 = \frac{1}{49.5} \times 100 \approx 2.02\% \]

### Final Answer
The second runner must increase his speed by approximately **2.02%** to win the race.

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