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Two towns $\mathrm{X}$ and $\mathrm{Y}$ are connected by a regular bus service. A bus leaves in either direction at every $t=T$ minutes. A man moving with some speed in the direction $\mathrm{X}$ to $\mathrm{Y}$ finds that a bus goes past him every $t=t_1$ minutes in the direction of his motion, and every $t=t_2$ minutes in the opposite direction. Then $T$ is given by
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Verified Answer
The correct answer is:
$\frac{2 t_1 t_2}{t_1+t_2}$
Let $\mathrm{V}$ be the speed of bus running between towns $\mathrm{A}$ and $\mathrm{B}$. And, speed of the man $=\mathrm{V}_{\mathrm{o}}$ Relative speed of the bus moving in the direction of man $=\mathrm{V}-\mathrm{V}_{\mathrm{o}}$
As bus went pass the man every $t_1$ minutes.
Since one bus leaves after every $T$ minutes, the distance
Similarly for bus moving in opposite direction.
$$
\begin{array}{ll}
\left(\mathrm{V}+\mathrm{V}_{\mathrm{o}}\right) \mathrm{t}_2=\mathrm{VT} & \quad \ldots \text { (iv) } \\
& \mathrm{So},\left(\mathrm{V}+\mathrm{V}_{\mathrm{o}}\right) \mathrm{t}_2=\left(\mathrm{V}-\mathrm{V}_{\mathrm{o}}\right) \mathrm{t}_1 \quad \text { [from (iii) \& (iv)] } \\
\Rightarrow \quad \mathrm{V}\left(\mathrm{t}_2-\mathrm{t}_1\right)=-\mathrm{V}_{\mathrm{o}}\left(\mathrm{t}_1+\mathrm{t}_2\right) & \\
\Rightarrow \quad \mathrm{V}=\frac{\mathrm{V}_{\mathrm{o}}\left(\mathrm{t}_1+\mathrm{t}_2\right)}{\mathrm{t}_1-\mathrm{t}_2} \\
\text { As, }\left(\mathrm{V}-\mathrm{V}_{\mathrm{o}}\right) \mathrm{t}_1=\mathrm{VT} \\
\Rightarrow \quad\left(1-\frac{\mathrm{V}_{\mathrm{o}}}{\mathrm{V}}\right) \mathrm{t}_1=\mathrm{T} \Rightarrow \mathrm{T}=\left(1-\frac{\mathrm{t}_1-\mathrm{t}_2}{\mathrm{t}_1+\mathrm{t}_2}\right) \mathrm{t}_1 \\
\Rightarrow \quad \mathrm{T}=\frac{2 \mathrm{t}_1 \mathrm{t}_2}{\mathrm{t}_1+\mathrm{t}_2}
\end{array}
$$
As bus went pass the man every $t_1$ minutes.
Since one bus leaves after every $T$ minutes, the distance
Similarly for bus moving in opposite direction.
$$
\begin{array}{ll}
\left(\mathrm{V}+\mathrm{V}_{\mathrm{o}}\right) \mathrm{t}_2=\mathrm{VT} & \quad \ldots \text { (iv) } \\
& \mathrm{So},\left(\mathrm{V}+\mathrm{V}_{\mathrm{o}}\right) \mathrm{t}_2=\left(\mathrm{V}-\mathrm{V}_{\mathrm{o}}\right) \mathrm{t}_1 \quad \text { [from (iii) \& (iv)] } \\
\Rightarrow \quad \mathrm{V}\left(\mathrm{t}_2-\mathrm{t}_1\right)=-\mathrm{V}_{\mathrm{o}}\left(\mathrm{t}_1+\mathrm{t}_2\right) & \\
\Rightarrow \quad \mathrm{V}=\frac{\mathrm{V}_{\mathrm{o}}\left(\mathrm{t}_1+\mathrm{t}_2\right)}{\mathrm{t}_1-\mathrm{t}_2} \\
\text { As, }\left(\mathrm{V}-\mathrm{V}_{\mathrm{o}}\right) \mathrm{t}_1=\mathrm{VT} \\
\Rightarrow \quad\left(1-\frac{\mathrm{V}_{\mathrm{o}}}{\mathrm{V}}\right) \mathrm{t}_1=\mathrm{T} \Rightarrow \mathrm{T}=\left(1-\frac{\mathrm{t}_1-\mathrm{t}_2}{\mathrm{t}_1+\mathrm{t}_2}\right) \mathrm{t}_1 \\
\Rightarrow \quad \mathrm{T}=\frac{2 \mathrm{t}_1 \mathrm{t}_2}{\mathrm{t}_1+\mathrm{t}_2}
\end{array}
$$
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