Problem 69 - Physics for Scientists and Engineers with Modern Physics

Question

(II) A motorboat whose speed in still water is 3.40 \(\mathrm{m} / \mathrm{s}\) must aim upstream at an angle of \(19.5^{\circ}\) (with respect to a line perpendicular to the shore) in order to travel dircctly across the stream. (a) What is the spced of the current? (b) What is the resultant speed of the boat with respect to the shore?

Step-by-Step Solution

Answer

Current speed: \(1.14 \, \mathrm{m/s}\); Resultant boat speed: \(3.21 \, \mathrm{m/s}\).

Step 1: Understand the Problem

The motorboat aims upstream at an angle of \( 19.5^{\circ} \) to travel straight across a river. The given speed of the boat in still water is \( 3.40 \ \mathrm{m/s} \). We need to find the speed of the river current (part a) and the resultant speed of the boat with respect to the shore (part b).

Step 2: Analyze the Components of Velocity

The velocity of the boat can be split into two components: one perpendicular to the shore and one parallel. The perpendicular component (\( V_{\perp} \)) equals \(V_b \cos(19.5^{\circ}) \), and the parallel component (\( V_{\parallel} \)) equals \(V_b \sin(19.5^{\circ}) \). Here, \( V_b = 3.40 \ \mathrm{m/s} \).

Step 3: Calculate the Speed of the Current

Since the boat travels directly across the stream, the parallel component of the boat's speed (\( V_{\parallel} \)) is equal to the speed of the current (\( V_c \)). Thus, \( V_c = 3.40 \sin(19.5^{\circ}) \).Calculate \( V_c \): \[V_c = 3.40 \sin(19.5^{\circ}) \approx 1.14 \ \mathrm{m/s}\]So, the current speed is approximately \( 1.14 \ \mathrm{m/s} \).

Step 4: Calculate the Resultant Speed of the Boat

The resultant speed of the boat with respect to the shore is just the perpendicular component since the parallel component (the speed of the current) only acts along the direction parallel to the shore. Therefore, \( V_{\text{resultant}} = V_{\perp} = V_b \cos(19.5^{\circ}) \).Calculate \( V_{\text{resultant}} \):\[V_{\text{resultant}} = 3.40 \cos(19.5^{\circ}) \approx 3.21 \ \mathrm{m/s}\]Thus, the resultant speed of the boat with respect to the shore is approximately \( 3.21 \ \mathrm{m/s} \).

Step 5: Summary of the Results

We found the speed of the current to be approximately \( 1.14 \ \mathrm{m/s} \) and the resultant speed of the boat with respect to the shore to be approximately \( 3.21 \ \mathrm{m/s} \).

Key Concepts

Velocity Components

When dealing with vector analysis, especially with objects moving in fluids like water or air, breaking down the velocity into components is key. This allows us to understand the pure movements along different directions. In our boat scenario, we only have two primary directions: the direction across the river (perpendicular to the shore) and the direction along the river (parallel to the current).
- **Perpendicular Component**: This is the velocity of the boat across the river. We calculate this using the cosine function due to how angles and velocities relate in the problem. The formula is given by \[ V_{\perp} = V_b \cos(\theta) \] where \( \theta \) is the angle made with the line perpendicular to the shore.
- **Parallel Component**: This controls how much the boat drifts downstream due to the current. It's calculated with the sine function: \[ V_{\parallel} = V_b \sin(\theta) \] Understanding these components is crucial, as they help us find how external factors like currents influence the final movement. By knowing the angle and speed in still water, these components help separate the speed purely across the river from the speed that's caused by the current.

Relative Motion

Relative motion deals with how an object moves in respect to another. For the motorboat, its movement needs to be analyzed about the shore and the water current.
- **Motion with Respect to the Water**: The boat can move at 3.40 m/s in still water. However, when we factor in the water's movement, things get interesting.
- **Motion with Respect to the Shore**: The boat's resultant velocity from the shore's perspective depends on its perpendicular component since the parallel component (due to the current) is canceled out when aiming to go straight across. In essence, understanding relative motion helps us explain why and how an object moves in a specific manner in real-world scenarios. Relative to the shore, the boat travels faster if there were no current! But in the problem, the boat's speed along the river's width is modified solely by its perpendicular component.

Trigonometric Functions

Trigonometry enables us to relate angles and lengths in our analysis of motion, especially within vector analysis.
- **Cosine Function**: Useful for finding adjacent (perpendicular) sides in right-angled triangles, representing the direct movement across the river in our scenario. Formula: \[ \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} \] Thus, the perpendicular component's formula becomes \( V_{\perp} = V_b \cos(19.5^\circ) \).
- **Sine Function**: Essential for the opposite side calculations, it relates to how much the current affects the boat. Formula: \[ \sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} \] Here, it's used to calculate the parallel component: \( V_{\parallel} = V_b \sin(19.5^\circ) \). These trigonometric functions allow us to parse out the complexities of motion, making it easier to understand and predict movement patterns when angles and forces come into play.