Question

In a race for $100m$ 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.

• A. $2\%$
• B. $4\%$
• C. more than $4\%$
• D. less than $4\%$

Answer 1

Answer: (C)

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 $50m$. During this time, the second runner has covered a distance $=50-1=49m$
So, $t=\frac{50}{v_1}=\frac{49}{v_2}$ .......(i)
Suppose, the second runner increases his speed to $v_3$ so that he covers the remaining distance $(=51m)$ in time $t$.
So $t=\frac{51}{v_3}=\frac{49}{v_2}$
or $v_3=\frac{51}{49}{v}_2$
Or $v_3=\left(1+\frac{2}{49}\right)v_2$
or $\frac{v_3}{v_2}-1=\frac{2}{49}$
or $\frac{v_3-v_2}{v_2}=\frac{2}{49}$
or $\%$ increase $=\frac{2}{49}\times 100=4.1\%$