Problem 83 - Fundamentals of Physics

Question

A woman who can row a boat at \(6.4 \mathrm{~km} / \mathrm{h}\) in still water faces a long, straight river with a width of \(6.4 \mathrm{~km}\) and a current of \(3.2 \mathrm{~km} / \mathrm{h}\). Let \(\hat{\text { i }}\) point directly across the river and \(\hat{\mathrm{j}}\) point directly downstream. If she rows in a straight line to a point directly opposite her starting position, (a) at what angle to î must she point the boat and (b) how long will she take? (c) How long will she take if, instead, she rows \(3.2 \mathrm{~km}\) down the river and then back to her starting point? (d) How long if she rows \(3.2 \mathrm{~km} u p\) the river and then back to her starting point? (e) At what angle to \(\hat{\mathbf{i}}\) should she point the boat if she wants to cross the river in the shortest possible time? (f) How long is that shortest time?

Step-by-Step Solution

Answer

(a) 30°, (b) 1.15 hours, (c) 1.33 hours, (d) 1.33 hours, (e) 0°, (f) 1 hour.

Step 1: Determine the required angle for direct crossing

The current is moving downstream at a speed of 3.2 km/h. To ensure that the woman reaches the opposite side directly, her velocity component downstream should equal the current. Her resultant velocity across the river is 6.4 km/h (the boat's speed in still water). To find the angle \( \theta \) she should row at, we set up the equation: \( 6.4 \sin(\theta) = 3.2 \). Solving for \( \theta \), we get: \( \sin(\theta) = \frac{3.2}{6.4} = 0.5 \), hence \( \theta = 30^\circ \).

Step 2: Calculate time to cross directly

The woman's velocity component directly across the river (i.e., perpendicular to the current) is given by \( 6.4 \cos(30^\circ) \). This calculates to \( 6.4 \cdot \frac{\sqrt{3}}{2} = 5.55 \) km/h. Since the river is 6.4 km wide, the time \( t \) to cross is \( \frac{6.4}{5.55} \approx 1.15 \) hours.

Step 3: Calculate time for downstream trip and return

For the downstream and return trip of 3.2 km each, the downstream velocity is the sum of her rowing speed and the current: \( 6.4 + 3.2 = 9.6 \) km/h. Upstream, her speed is the rowing speed minus the current: \( 6.4 - 3.2 = 3.2 \) km/h. Downstream time is \( \frac{3.2}{9.6} = \frac{1}{3} \) hours, and upstream time is \( \frac{3.2}{3.2} = 1 \) hour. Total time is \( \frac{1}{3} + 1 \approx 1.33 \) hours.

Step 4: Calculate time for upstream trip and return

For an upstream and return trip of 3.2 km each, the calculation is the same as the downstream trip in reverse. Upstream time is \( \frac{3.2}{3.2} = 1 \) hour, and downstream return time is \( \frac{3.2}{9.6} = \frac{1}{3} \) hours. Total time is \( 1 + \frac{1}{3} = 1.33 \) hours.

Step 5: Determine angle for shortest crossing time

To cross the river in the shortest time, the woman should aim directly across the river, perpendicular to the current. This means the angle \( \theta \) with the vector \( \hat{i} \) is \( 0^\circ \).

Step 6: Calculate shortest crossing time

Rowing directly across, her speed in the direction across the river remains the full speed 6.4 km/h. The river’s width is 6.4 km, so the shortest time\( t \) is \( \frac{6.4}{6.4} = 1 \) hour.

Key Concepts

River Crossing Problems

River crossing problems in physics involve understanding how an object moves across a body of water with currents. These problems require calculating the resultant path when influenced by both the object's velocity and the river's current speed. Understanding this type of motion is crucial in solving problems related to crossing any moving fluid. - The primary challenge involves directing the object, such as a boat, in a manner that compensates for the river's current. - One must determine the necessary angle to row or steer in order to reach a certain point on the opposite bank. - It is often necessary to find the shortest or most efficient route across.
For the problem at hand, the woman rowing needs to calculate the angle to offset the river's current, allowing her to reach her target point on the other side directly.

Vector Analysis

When solving a river crossing problem, vector analysis is a powerful tool. A vector represents a quantity having both magnitude and direction. Understanding vectors is key to handling the two-dimensional movement in this scenario.- The boat's velocity can be broken down into two components: one directly across the river (indicated by \(\hat{i}\)) and one parallel to the river's downstream direction (indicated by \(\hat{j}\)).- To determine the actual path, one constructs a vector triangle where the hypotenuse is the resultant velocity.- Solving for components using trigonometry often involves equations like \( 6.4 \sin(\theta) = 3.2 \), where \(\theta\) is the angle the boat should be oriented relative to \(\hat{i}\).This analysis allows for precise calculation of the boat's movement relative to the water, ensuring it reaches the desired destination despite external influences.

Relative Velocity

Relative velocity takes into account the movement of an object with respect to another moving object. In the context of river crossing, it’s about understanding how the boat's speed in still water combines with the river’s current. - The relative velocity is the resultant of both the boat's speed and the river's flow. - By comparing the velocity component in the direction perpendicular to the current, one determines the effective speed across the river. This is critical for calculating crossing time. - For instance, the woman's rowing speed and the river's speed create a situation where she must row at 30° to counteract the downstream push. Solving these relative motion problems often provides insights into how adjustments in direction affect overall travel time across the river.

Trigonometry in Physics

Trigonometry is frequently employed in physics to analyze motion involving angles. This mathematical concept plays a crucial role in river crossing problems, where determining angles is essential.- By using trigonometric functions like sine and cosine, one can resolve vectors into perpendicular components.- For example, the equation \( \sin(\theta) = \frac{3.2}{6.4} \) comes into play when solving for the necessary angle \(\theta\) to neutralize the current.- It further helps in finding the effective component speed across the river using formulas like \( 6.4 \cos(\theta) \).
Understanding how to apply trigonometry in physics helps simplify complex real-world problems, such as determining the path of optimal travel across a moving body of water.