Question

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.

• 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 $50 \,m$. During this time, the second runner has covered a distance $=50-1=49 \,m$
So, $t=\frac{50}{v_{1}}=\frac{49}{v_{2}}$ .......(i)
Suppose, the second runner increases his speed to $v_{3}$ so that he covers the remaining distance $(=51 \,m )$ 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 $=\frac{2}{49} \times 100=4.1 \%$