Problem 76 - College Physics

Question

A boat that travels at a speed of \(6.75 \mathrm{~m} / \mathrm{s}\) in still water is to go directly across a river and back ( \(>\) Fig. 3.38 ). The current flows at \(0.50 \mathrm{~m} / \mathrm{s}\). (a) At what angle(s) must the boat be steered? (b) How long does it take to make the round trip? (Assume that the boat's speed is constant at all times, and neglect turnaround time.)

Step-by-Step Solution

Answer

Steer the boat at approximately \(4.25^\circ\) upstream. Calculate the round trip time using the river width.

Step 1: Determine the Angle to Counteract the Current

To travel straight across the river, the boat must be steered at an angle \( \theta \) against the current. The component of the boat's velocity in the direction opposite to the current must equal the current's velocity to cancel it.\[V_{b} \sin \theta = V_{c}\]Where \(V_{b}=6.75\) m/s is the boat's speed and \(V_{c}=0.50\) m/s is the current speed. Solving for \(\theta\),\[\sin \theta = \frac{0.50}{6.75} = 0.0741\]\[\theta = \arcsin(0.0741) \approx 4.25^\circ\]The boat must be steered approximately at an angle of \(4.25^\circ\) upstream against the current.

Step 2: Calculate the Perpendicular Speed

Once the correct angle is set, determine the component of the boat's velocity that is perpendicular to the current, aiding in crossing the river. This is calculated as:\[V_{b\perp} = V_{b} \cos \theta\]\[V_{b\perp} = 6.75 \cos(4.25^\circ) \approx 6.74 \text{ m/s}\]This is the effective speed of the boat directly across the river.

Step 3: Determine the Round Trip Time

The time taken to cross one way at speed \(V_{b\perp}\) is given by distance divided by speed. Let \(d\) be the width of the river, then the time to cross one way is\[T_{one\,way} = \frac{d}{V_{b\perp}}\]Thus, the round trip time is\[T_{round\,trip} = 2 \times \frac{d}{6.74}\]Without the distance \(d\), the calculation cannot be completed numerically, but the formula is prepared for the given dimensions of the river.

Key Concepts

Trigonometry in Physics

Trigonometry often steps in to solve physics problems involving angles and distances, making it an essential tool for understanding the behavior of objects in motion. In our case of the river crossing problem, we employ trigonometry to determine the angle the boat must be steered to counteract the river's current.
The problem involved using the sine of the angle. This was necessary to ensure that the boat's sideways velocity component fully negates the water current's speed. We have seen the equation:

• \(\sin \theta = \frac{0.50}{6.75}\)

Our work paid off when we calculated \(\theta \approx 4.25^\circ\), showing how tiny trigonometric functions can impact real-world scenarios. Understanding these basic trigonometric principles is crucial for students as it lays the groundwork for tackling more complex physics problems involving multiple forces and directions.

Relative Motion

The concept of relative motion is vital for solving problems where different speeds act simultaneously, like the river crossing problem. Here, we consider how the boat's motion is influenced by the river's flow.
In our example, the river's current moves at \(0.50 \text{ m/s}\) while the boat travels at \(6.75 \text{ m/s}\) in still water. The boat must counteract the flow to travel straight across. We established that the boat's velocity relative to the ground must combine with the river velocity to form the resultant motion.
Relative motion concludes that our calculated angle aligns the boat’s trajectory so its effective path counterbalances the river, steering the boat straight across. This concept is essential in physics, especially when describing motions from various perspectives.

Vector Components

Understanding vector components is key to managing complex motion in physics. Vectors allow us to break down forces into easily workable parts, like direction and magnitude.
In the river problem, the boat's speed can be dissected into two vector components:

• One parallel to the river current
• One perpendicular to the river current

Using trigonometry, we calculated the perpendicular velocity component, \(V_{b\perp} = 6.74 \text{ m/s}\), that determines how fast the boat moves straight across. This highlights how components simplify problems involving angles and multiple forces, guiding accurate calculations for unpredictable environments.

Physics Problem Solving Steps

Solving physics problems often involves clearly defined steps which simplify complex situations. Let's break down the river crossing problem as an example:

• Identify knowns and unknowns: We examined both the boat's and river's speeds.
• Use appropriate formulas: We applied trigonometric functions to decide the necessary angle.
• Break vectors into components: We calculated the components for straightforward motion.
• Derive further equations: Formulated for the round trip time, even when not fully solvable.

Each stage plays a crucial role, illustrating the systematic approach needed in physics problem-solving, which minimizes errors and enhances understanding. Mastery of these steps empowers students to tackle various physics challenges effectively.