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How would you solve this in detail (in English)?

The problem can be summarized as follows:

You are trying to steer a boat across a 50-meter-wide river. The boat has a speed of \( |\vec{v}| = 5 \, \text{m/s} \) in still water, and the river flows with a velocity \( |\vec{u}| = 2 \, \text{m/s} \). The goal is to find the direction in which you must steer the boat in order to reach the opposite ferry landing exactly across the river, and to calculate the time required for the crossing.

### Step-by-Step Solution:

#### 1. Analyze the velocities involved:
- The boat has a speed \( |\vec{v}| = 5 \, \text{m/s} \).
- The river current flows at a speed \( |\vec{u}| = 2 \, \text{m/s} \).
  
These are vector quantities. The boat’s velocity can be decomposed into two components:
- One component perpendicular to the flow of the river (in the direction across the river).
- One component parallel to the flow (to counteract the current).

#### 2. Geometry of the problem:
Let’s assume you steer the boat at an angle \( \theta \) with respect to the direction directly across the river.

- The boat’s velocity perpendicular to the river’s current must equal the boat’s speed across the river, which is \( v_y \).
- The river’s current will move the boat downstream if not countered. To arrive directly across from your starting point, you need the horizontal component \( v_x \) of the boat’s velocity to cancel out the flow of the river.

#### 3. Set up the components of velocity:
- The perpendicular component to the flow of the river (across the river): \( v_y = v \cdot \cos(\theta) \).
- The component along the flow of the river (to counter the current): \( v_x = v \cdot \sin(\theta) \).

To exactly counter the current, the horizontal component of the boat’s velocity must cancel out the current’s velocity:
\[
v_x = u \quad \Rightarrow \quad v \cdot \sin(\theta) = u
\]
\[
\sin(\theta) = \frac{u}{v} = \frac{2}{5} = 0.4
\]
\[
\theta = \arcsin(0.4) \approx 23.58^\circ
\]

So, the boat must be steered at an angle of approximately \( 23.58^\circ \) upstream relative to the direction across the river.

#### 4. Calculate the time taken to cross the river:
The speed of the boat across the river (in the direction perpendicular to the flow) is given by the perpendicular component:
\[
v_y = v \cdot \cos(\theta)
\]
We already know \( \theta = 23.58^\circ \), so:
\[
v_y = 5 \cdot \cos(23.58^\circ) \approx 5 \cdot 0.92 = 4.6 \, \text{m/s}
\]

The time to cross the river is then:
\[
t = \frac{\text{width of river}}{v_y} = \frac{50 \, \text{m}}{4.6 \, \text{m/s}} \approx 10.87 \, \text{seconds}
\]

### Final Answer:
- **Direction**: You must steer the boat at an angle of approximately \( 23.58^\circ \) upstream relative to the direct path across the river.
- **Time**: The crossing will take approximately \( 10.87 \, \text{seconds} \).

---

Do you want any further details or have any questions?  
Here are five related questions to explore further:
1. How would the crossing time change if the river current were stronger?
2. What would happen if the boat's speed were slower than the river current?
3. How can we calculate the boat's velocity relative to the riverbank?
4. What would the angle \( \theta \) be if the boat were faster?
5. How could you solve this problem graphically using vector addition?

**Tip**: Always decompose vectors into components parallel and perpendicular to a reference direction when solving motion problems in multiple dimensions.

You are trying to steer a boat across a 50-meter-wide river. The boat has a speed of 5 m/s, and the river flows with a velocity of 2 m/s. In which direction must you steer to arrive directly opposite, and how long will it take to cross?

Learn how to calculate the direction to steer a boat with a speed of 5 m/s across a 50-meter-wide river with a current of 2 m/s. Find the angle needed to counter the current and the time required for the crossing. This solution covers vector decomposition and trigonometric calculations.

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