Question

A person starts from rest and accelerates uniformly for the first $50m$ of a $100m$ race, and then continues at a constant velocity for the second $50m$ of the race. If the person completes the $100m$ in a time of $10s$, Find out his instantaneous velocity when he crosses the finish line?

• A. $5m/s$
• B. $10m/s$
• C. $12m/s$
• D. $15m/s$
• E. $20m/s$

Answer 1

The correct option is D $15m/s$

Given : Total time taken $T=10$ s $u=0$ m/s

Let the time taken during 1st half and second half be $t_1$ and $t_2$ respectively.

$\therefore$ $T=t_1+T_2$ $⟹t_2=10-t_1$

Let the acceleration of the person during 1st half be $a$.

For 1st half : $S=ut+\frac{1}{2}a{t}^2$

$\therefore$ $50=0+\frac{1}{2}a{t}_1^2$ $⟹a=\frac{100}{t_1^2}$ ...............(1)

Also $V_1^2-0^2=2a\left(50\right)$ $⟹V_1^2=100a$ ..........(2)

2nd half : $50=V_1{t}_2=V_1(10-t_2)$

$⟹$ $V_1=\frac{50}{10-t_1}$ ..............(3)

Using (1) and (3) in (2) we get $\frac{(50{)}^2}{(10-t_1{)}^2}=100\times \frac{100}{t_1^2}$

OR $\frac{t_1}{10-t_1}=2$ $⟹t_1=\frac{20}{3}$ s

From (3), $V_1=\frac{50}{10-\frac{20}{3}}=15$ $m/s$