The current velocity of river grows in proportion to the distance from its bank and reaches the maximum value `v_0` in the middle. Near the banks the velocity is zero. A boat is moving along the river in such a manner that the boatman rows his boat always perpendicular to the current. The speed of the boat in still water is u. Find the distance through which the boat crossing the river will be carried away by the current, if the width of the river is c. Also determine the trajectory of the boat.

Explanation

To solve the problem, we will break it down into steps:

### Step 1: Understand the velocity of the river
The current velocity of the river varies with the distance from the bank. It is zero at the banks and reaches a maximum value \( v_0 \) in the middle of the river. This means that if we consider a point at a distance \( y \) from one bank, the velocity of the river at that point can be expressed as:
\[ v(y) = \frac{v_0}{\frac{c}{2}} \cdot y = \frac{2v_0}{c} \cdot y \]
where \( c \) is the width of the river.

### Step 2: Determine the effective velocity of the boat
The boatman rows the boat perpendicular to the current. The speed of the boat in still water is \( u \). The effective velocity of the boat with respect to the ground (considering both the boat's speed and the river's current) can be calculated using the Pythagorean theorem:
\[ v = \sqrt{u^2 + v^2} \]
where \( v \) is the velocity of the river at the midpoint (which is \( v_0 \) when the boat is in the middle).

### Step 3: Calculate the time taken to cross the river
The time taken \( t \) to cross the river of width \( c \) can be calculated as:
\[ t = \frac{c}{u} \]
This is because the boat is moving at a speed \( u \) perpendicular to the current.

### Step 4: Calculate the distance carried away by the current
During the time \( t \), the current will carry the boat downstream. The distance \( d \) carried away by the current can be calculated as:
\[ d = v_0 \cdot t \]
Substituting \( t \) from step 3:
\[ d = v_0 \cdot \frac{c}{u} \]

### Step 5: Determine the trajectory of the boat
The trajectory of the boat can be determined by the angle \( \theta \) it makes with the bank. This angle can be found using:
\[ \tan(\theta) = \frac{v_0}{u} \]
Thus, the angle \( \theta \) is:
\[ \theta = \tan^{-1}\left(\frac{v_0}{u}\right) \]

### Final Formulas
1. Distance carried away by the current:
\[ d = \frac{v_0 \cdot c}{u} \]
2. Trajectory angle:
\[ \theta = \tan^{-1}\left(\frac{v_0}{u}\right) \]