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Relative Velocity in 2D

A boat takes ...

Question

A boat takes $t_{1}$ time while going downstream between two fixed points $A$ and $B$ , and takes $t_{2}$ time while going upstream through the same distance. If river does not flow, then the time taken by the boat to cover the same distance is

t = \frac{2 t_{1} t_{2}}{t_{1} + t_{2}}

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t = \frac{t_{1} t_{2}}{t_{1} + t_{2}}

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t = \frac{2 t_{2}}{t_{1} + t_{2}}

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t = \frac{2 t_{1}}{t_{1} + t_{2}}

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Solution

The correct option is A $t = \frac{2 t_{1} t_{2}}{t_{1} + t_{2}}$

Let $V_{b}$ be the velocity of the boat in still water and $V_{R}$ be the flow speed in the river.

For downstream:
$V_{b} + V_{R} = V_{r e l}$
$\Rightarrow V_{b} + V_{R} = \frac{x}{t_{1}} . . . . . (1)$

For upstream:
$V_{b} - V_{R} = V_{r e l}$
$\Rightarrow V_{b} - V_{R} = \frac{x}{t_{2}} . . . . . . (2)$

Adding equations $(1)$ and $(2)$ , we get:
$2 V_{b} = \frac{x}{t_{1}} + \frac{x}{t_{2}}$
$2 V_{b} = x (\frac{t_{1} + t_{2}}{t_{1} . t_{2}})$
but, $\frac{x}{V_{b}} = t$
$\Rightarrow t = \frac{x}{V_{b}} = \frac{2 t_{1} t_{2}}{t_{1} + t_{2}}$

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