To understand the path of the boat as it crosses the river while being affected by the current, we need to analyze the situation using some basic principles of physics and geometry. The boat is rowed with a constant velocity \( u \) towards a point directly opposite its starting point, while the river flows with a velocity \( \nu u \). This creates an interesting dynamic where the boat's path will not be a straight line but rather a curve due to the influence of the river's current.
Setting Up the Problem
Let's define the variables clearly:
- u: Velocity of the boat relative to the water.
- ν: The ratio of the river's current speed to the boat's speed (i.e., the current speed is \( \nu u \)).
- d: Width of the river.
- θ: The angle at which the boat is pointed relative to the bank.
Understanding the Motion
The boat is always directed towards point O, which is directly across from point A. However, because of the river's current, the boat will drift downstream as it moves across the river. To find the path equation \( r = f(\theta) \), we need to consider the components of the boat's velocity.
Velocity Components
The boat's velocity can be broken down into two components:
- The component perpendicular to the river bank (across the river): \( u \cos(\theta) \)
- The component parallel to the river bank (downstream): \( u \sin(\theta) \)
The river's current adds an additional velocity component of \( \nu u \) downstream. Therefore, the total downstream velocity of the boat becomes:
Vdownstream = u \sin(\theta) + \nu u
Time to Cross the River
The time \( t \) it takes for the boat to cross the river can be calculated using the width \( d \) and the perpendicular component of the boat's velocity:
t = \frac{d}{u \cos(\theta)}
Displacement Downstream
During this time, the boat will also drift downstream due to both its own motion and the current. The total downstream displacement \( x \) can be expressed as:
x = (u \sin(\theta) + \nu u) t
Substituting for \( t \), we get:
x = (u \sin(\theta) + \nu u) \frac{d}{u \cos(\theta)}
After simplifying, this becomes:
x = d \left( \tan(\theta) + \nu \right)
Path Equation
Now, we can express the relationship between the downstream displacement \( x \) and the width of the river \( d \) in terms of the angle \( \theta \). Rearranging gives us the path equation:
r = d \left( \tan(\theta) + \nu \right)
Visualizing the Path
This equation shows that the path of the boat is a function of the angle \( \theta \) at which it is pointed. As the angle changes, the path will curve, illustrating how the boat is affected by both its own rowing and the river's current. The greater the angle \( \theta \), the more pronounced the downstream drift will be.
Conclusion
In summary, the path of the boat across the river can be described by the equation \( r = d \left( \tan(\theta) + \nu \right) \). This relationship highlights the interplay between the boat's velocity and the river's current, resulting in a curved trajectory rather than a straight line. Understanding this dynamic is crucial for navigating effectively in flowing water.
