1. River mechanics question

I'm having trouble with this question and I appreciate anyone's help.

A man swims across a straight river of uniform width W, starting from a point O on one bank of the river. The velocity of the river at a distance y from the bank is
u(y)=ay(W − y), where a is a positive constant. The man travels at a constant speed v relative to the current and steers a course set at a
constant angle θ (0 < θ < pi) to the downstream direction.

(a) Show that the velocity of the man relative to a Cartesian coordinate system with origin at O, i pointing in the downstream direction, and j pointing across the river is given by (u + v cos θ)i + (v sin θ)j. I've done this part.

(b) At what time does the man reach the other bank?
I get t = W/v.sinθ - I think that's correct.

(c) Show that when the man has reached the other bank, the downstream distance he has travelled is equal to aW^3/6v.sinθ + W.cot θ.

This is the part i'm having problems with. I thought about replacing t from part (b) into the integrated i component from the velocity in part (a). I'm not sure what to do with the u(y) though.

2. Originally Posted by 0-)
I'm having trouble with this question and I appreciate anyone's help.

A man swims across a straight river of uniform width W, starting from a point O on one bank of the river. The velocity of the river at a distance y from the bank is
u(y)=ay(W − y), where a is a positive constant. The man travels at a constant speed v relative to the current and steers a course set at a
constant angle θ (0 < θ < pi) to the downstream direction.

(a) Show that the velocity of the man relative to a Cartesian coordinate system with origin at O, i pointing in the downstream direction, and j pointing across the river is given by (u + v cos θ)i + (v sin θ)j. I've done this part.

(b) At what time does the man reach the other bank?
I get t = W/v.sinθ - I think that's correct.

(c) Show that when the man has reached the other bank, the downstream distance he has travelled is equal to aW^3/6v.sinθ + W.cot θ.

This is the part i'm having problems with. I thought about replacing t from part (b) into the integrated i component from the velocity in part (a). I'm not sure what to do with the u(y) though.
The swimmer's speed in the j direction is constant (v sin θ). So at time t, when he has travelled a distance y in that direction, y will be equal to (v sin θ)t. For that value of y, the corresponding value of u will be a(v sin θ)t(W – (v sin θ)t).

To get the total distance travelled downstream, we have to integrate the velocity u + v cos θ as t goes from 0 to W/v.sin θ:

. . . . . .Distance $\displaystyle = \int_0^{W/v\sin\theta} \!\!\!\!\!\!\bigl(av\sin\theta.t(W - v\sin\theta.t) + v\cos\theta\bigr)dt$.

Integrate that and you should get the right answer.