A river of width 'd' with straight parallel banks flows due North with speed u. A boat, whose speed is v relative to water, starts from A and crosses the river. If the boat is steered due West and u varies with y as `u=(y(d-y)v)/d^(2)` then answer the following questions.

Equation of trajectory of the boat is

Options

A. `y=x^(2)/(2d)`
B. `x=y^(2)/(2d)`
C. `y=x^(2)/(2d)-x^(3)/(3d^(2))`
D. `x=y^(2)/(2d)-y^(3)/(3d^(2))`

Answer

D

Explanation

For boat (w.r.t. ground) `v_(y)=v, v_(x)=u=(y(d-y))/d^(2)vrArr (dy)/(dt)=v` and `(dx)/(dt)=v=(y(d-y))/d^(2) v`
`rArr (dx)/(dy)=(y(d-y))/d^(2)rArr int_(0)^(x)dx=int_(0)^(y)((yd-y^(2)))/d^(2) dy rArr x=y^(2)/(2d)-y^(3)/(3d^(2))`