Question

A length of path $A C B$ is $1500\, m$ and the length of the path $A D B$ is $2100\, m$. Two particles start from point $A$ simultaneously around the track $A C B D A$. One of them travels the track in clockwise sense and other in anticlockwise sense with their respective constant speeds. After $12\, s$ from the start, the first time they meet at the point $B$. After minimum time(ins) in which they meet first at point $B$, will they again meet at the point $B$ is time $t_{\min }=(12)^{x} s$. The value of $x$ is______.

Answer 1

The velocity of first particle is $v_{1}=\frac{1500}{12} ms ^{-1}$ and the velocity of second particle is $v_{2}=\frac{2100}{12} ms ^{-1}$.
Let after completing $n_{1}$ and $n_{2}$ trips, they will again meet at the point $B$.
$\therefore \frac{3600 n_{1}}{v_{1}}=\frac{3600 n_{2}}{v_{2}}=t$
Or $\frac{n_{1}}{n_{2}}=\frac{v_{1}}{v_{2}}=\frac{\frac{1500}{\frac{12}{2100}}}{\frac{12}{2}}$
$=\frac{1500}{2100}=\frac{5}{7}$
$\therefore t_{\min }=\frac{3600 \times 5}{v_{1}}$
$=\frac{3600 \times 5}{1500} \times 12=144\, s =(12)^{x}$
$\therefore x=2$