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A man wants to swim in a river from A to...

A man wants to swim in a river from `A to B` and back from `B to A` always following line `AB` (Fig. 5.94). The distance between points `A and B` is `S`. The velocity of the river current `v` is constant over the entire width of the river. The line `AB` makes an angle `prop` with the direction of current. The man moves with velocity `u` at angle `beta` to the line `AB`. The man swim to cover distance `AB` and back, find the time taken to complete the journey.
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Text Solution

Verified by Experts

In this problem, we choose axis along `AB` and normal to is it.
Since the man moves along `AB`, the velocity components of current and man normal to `AB` must cancel out, i.e.,
`u sin beta = v sin prop`
When the man moves from `A` to `B`, his resultant velocity along `AB = (u cos beta + v cos prop)`
Hence, `S = (u cos beta + v cos prop) t_1`
while for motion from `B` to `A S = (u cos beta - v cos prop)t_2`
From the condition of the problem, `t_1 + t_2 = t`
`(S)/(u cos beta + v cos prop) + (S)/( u cos beta - v cos prop) = t`
`s [(u cos beta - v cos prop + u cos beta + v cos prop)/(u^2 cos^2 beta - v^2 cos^2 prop)] = t`
`(S(2 u cos beta))/(u^2 cos^2 beta - v^2 cos^2 prop) = t`.
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