In a race for 100 m dash, the first and the second runners have a gap of one metre at the mid way stage. Assuming the first runner goes steady, by what percentage should the second runner increases his speed just to win the race.

Let $v_1 and v_2$ be the initial speeds of first and second runners respectively. Let t be time by them when the first runner has completed 50 m. During this time, the second runner has covered a distance = 50 -1= 49 m

So, $t=\frac{50}{v_1}=\frac{49}{v_2}..............\left(i\right)$

Suppose, the second runner increases his speed to $v_3$ so that he covers the remaining distance ( = 51m) in time t.

So, $t=\frac{51}{v_3}=\frac{49}{v_2}$

So, $v_3=\frac{51}{49}{v}_2$

So $v_3=\left(1+\frac{2}{49}\right)v_2$

So, $\frac{v_3}{v_2}-1=\frac{2}{49}$

So, $\frac{v_3-v_2}{v_2}=\frac{2}{49}$

Or % increase $=\frac{2}{49}\times 100=4.1\%$