Problem

A river of width

a

with straight parallel banks flows due north with speed

u

. The points

O

and

A

are on opposite banks and

A

is due east of

O

. Coordinate axes

Ox

and

Oy

are taken in the east and north directions respectively. A boat, whose speed is

v

relative to water, starts from

O

and crosses the river. If the boat is steered due east and

u

varies with

x

as :

u = x(a - x) \frac{v}{a^2}

. Find a. equation of trajectory of the boat, b. time taken to cross the river, c. absolute velocity of boatman when he reaches the opposite bank, d. the displacement of boatman when he reaches the opposite bank from the initial position.

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