A rowboat crosses a river with a velocity of 3.36 mi/h at an angle 62.5° north of west relative to the water. The river is 0.595 mi wide and carries an eastward current of 1.25 mi/h. How far upstream is the boat when it reaches the opposite shore?

To solve this problem, we need to break down the velocity of the boat into its components relative to the riverbank and then calculate how far upstream it will be when it reaches the opposite shore. Here's a step-by-step breakdown of the solution:

### Step 1: Break the boat's velocity into components
The boat's velocity relative to the water is given as 3.36 mi/h at an angle of 62.5° north of west. To analyze the motion, we decompose this velocity into its westward (horizontal) and northward (vertical) components.

- **Westward component**: 
  \[
  v_{\text{west}} = 3.36 \times \cos(62.5^\circ)
  \]

- **Northward component**: 
  \[
  v_{\text{north}} = 3.36 \times \sin(62.5^\circ)
  \]

### Step 2: Calculate the boat's speed relative to the shore
The river has an eastward current of 1.25 mi/h. This affects the boat’s total velocity in the horizontal direction. The total horizontal velocity relative to the shore is the westward velocity minus the eastward current.

- **Net westward velocity**: 
  \[
  v_{\text{net west}} = v_{\text{west}} - 1.25
  \]

The northward velocity remains unchanged since the current only affects the east-west direction.

### Step 3: Time to cross the river
We are given that the river is 0.595 miles wide, and the boat's velocity across the river (northward) is \( v_{\text{north}} \). The time taken to cross the river is:

\[
t = \frac{\text{width of the river}}{v_{\text{north}}} = \frac{0.595}{v_{\text{north}}}
\]

### Step 4: Distance traveled upstream
The upstream distance is the total westward distance the boat moves while crossing the river. This can be calculated using:

\[
d_{\text{upstream}} = v_{\text{net west}} \times t
\]

Now, let's compute these values.Here are the results of the calculations:

1. **Westward velocity**: \(1.55 \, \text{mi/h}\)
2. **Northward velocity**: \(2.98 \, \text{mi/h}\)
3. **Net westward velocity** (after accounting for the eastward current): \(0.30 \, \text{mi/h}\)
4. **Time to cross the river**: \(0.20 \, \text{hours}\)
5. **Distance traveled upstream**: \(0.0602 \, \text{miles} \, \text{(or approximately 318 feet)}\)

Thus, the boat ends up approximately **0.0602 miles** (or 318 feet) upstream when it reaches the opposite shore.

Let me know if you'd like any additional details, or if you have more questions. Here are some related questions to explore:

1. What if the current speed increased by 0.5 mi/h?
2. How long would it take if the river was twice as wide?
3. What if the boat's velocity increased to 4.5 mi/h?
4. How would the angle need to change to ensure the boat reaches a specific point directly across the river?
5. How far downstream would the boat travel if the current was in the opposite direction?

**Tip**: Breaking vectors into components is key to solving many motion-related problems!

A rowboat crosses a river with a velocity of 3.36 mi/h at an angle 62.5° north of west relative to the water. The river is 0.595 mi wide and carries an eastward current of 1.25 mi/h. How far upstream is the boat when it reaches the opposite shore?

Solve the problem of a rowboat crossing a river with a velocity of 3.36 mi/h at an angle of 62.5° north of west. Learn how to decompose velocity, calculate net velocity, and determine how far upstream the boat will travel. Suitable for students in Grades 10-12.

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