A broad pipe having a radius `10 cm` branches into two pipes of radii, `5 cm` and `3cm`. If the velocity of flowing water in the pipe of radius `3cm` be `5cm//s`, determine the velocities of water in the remaining two pipes. Given that the rate of discharge through the main branch, is `600 pi cm^(3)//s`

Options

A. `v_(1)=6 cm//s and v_(2)=22.2 cm//s`
B. `v_(1)=4 cm//s and v_(2)=12.2 cm//s`
C. `v_(1)=4 cm//s and v_(2)=12.2 cm//s`
D. None of these

Answer

A

Explanation

Consider any three sections (1), (2) and (3) in the three pipes of different radii, as shown the figure, if `v_(1)` and `v_(2)` be the velocity of water at section (1) and (2) respectively, then

`v_(1) = , v_(2) = ? and v_(3) = 5 cm//s`.
The rate of discharge the three pipes are
`a_(1)v_(1) = 100 pi v_(1) cm^(3)//s , a_(2)v_(2) = 25piv_(2) cm^(3)//s`
and `a_(3)v_(3)= 9 pi xx 5 = 45pi cm^(3)//s`
Now, `Q = a_(1)v_(1) = a_(2)v_(2)+a_(3)v_(3)`
`implies 600 pi = 100 pi v_(1) = 25 pi v_(2)+45 pi`
Solving, we get , `v_(1) = 6 cm//s` and `v_(2) = 22.2 cm//s`.