Problem 80 - College Physics

Question

You are in a fast powerboat that is capable of a sustained steady speed of \(20.0 \mathrm{~m} / \mathrm{s}\) in still water. On a swift, straight section of a river you travel parallel to the bank of the river. You note that you take \(15.0 \mathrm{~s}\) to go between two trees on the riverbank that are \(400 \mathrm{~m}\) apart. (a) (1) Are you traveling with the current, (2) are you traveling against the current, or (3) is there no current? (b) If there is a current [reasoned in part (a)], determine its speed.

Step-by-Step Solution

Answer

(1) Traveling with the current. The current speed is 6.67 m/s.

Step 1: Understand the Problem

You are asked to determine whether you are traveling with the current, against the current, or if there's no current, based on the given travel time and distance between two points along a riverbank.

Step 2: Calculate Observed Speed

Determine the speed of the powerboat when measured over the specified distance. Given the distance is 400 meters and the time is 15 seconds, use the formula for speed: \[ v_{observed} = \frac{\text{distance}}{\text{time}} = \frac{400 \, \text{meters}}{15 \, \text{seconds}} = 26.67 \, \text{m/s} \]

Step 3: Compare with Boat Speed in Still Water

Compare the observed speed of 26.67 m/s to the boat's speed in still water (20.0 m/s). Since 26.67 m/s is greater than 20.0 m/s, the boat is traveling faster than it would in still water, indicating that the current is assisting the boat's motion.

Step 4: Determine Direction of Travel in Relation to Current

Since the observed speed is greater than the still water speed, you are traveling with the current. This means the current is in the same direction as the boat's movement.

Step 5: Calculate Current Speed

To find the speed of the current, subtract the boat's speed in still water from the observed speed. Use the following formula:\[ v_{current} = v_{observed} - v_{still\,water} = 26.67 \, \text{m/s} - 20.0 \, \text{m/s} = 6.67 \, \text{m/s} \]

Key Concepts

Physics

The physics behind the Riverboat Problem involves the concept of motion. Motion in physics usually describes a change in position of an object with respect to time. This encompasses a wide variety of topics such as velocity, speed, and acceleration.
In this problem, understanding motion and speed is crucial. Motion is defined by both a direction and a speed, and in the context of fluid dynamics, we examine how these relate to an object like a boat moving through water. The speed of the boat relative to the ground includes both the speed of the boat through the water and the speed of the water itself (the current).
The Riverboat Problem provides an applied context within physics that involves these key concepts, and serves as an insightful application of physics principles in real-world scenarios. Such problems help students gain a clearer understanding of how principles like forces, speeds, and streams interact with each other.

Kinematics

In the Riverboat Problem, kinematics plays a vital role in understanding how objects move without regard to the forces that cause them. Kinematics is the branch of mechanics that describes the motion of points, bodies, and systems of bodies.
Key elements include:

• Displacement: The measurement of the shortest distance between initial and final points. In the problem, the boat covers 400 meters.
• Speed and Velocity: Speed is a scalar quantity and measures the rate of motion distance, while velocity is a vector quantity including direction. The calculation of the boat's speed on the river serves as a practical example of these concepts.
• Time: This is a crucial element, as the problem defines a specific time duration (15 seconds) to analyze the boat's movement.

The given values allow us to calculate the observed speed and compare it against the speed in still water. This comparison helps determine the effect of the river’s current in the motion of the boat.

Relative Motion

Relative motion describes how the position of an object changes with respect to another moving object. In this problem, it highlights the importance of viewpoint; the boat's speed relative to the bank differs from its speed in still water.
In our scenario, we have:

• Boat's Speed in Still Water: This is given as 20 m/s, which is the speed of the boat relative to water, assuming no water motion.
• Observed Speed: This is the speed relative to the riverbank (or ground), calculated as 26.67 m/s.

The relative motion here is notable because it distinguishes between the boat’s resultant speed over the ground and its innate speed through the water alone. If the observed speed is greater than the still water speed, it tells us that another factor, the river's current, must be influencing the boat's motion. Understanding relative motion allows for identifying the presence and behavior of this additional factor.

Current Speed Calculation

Calculating the speed of the river's current is straightforward and involves simple arithmetic once you understand the concept of relative motion. By knowing the observed speed and the speed in still water, we can find the current’s influence.
When the observed speed exceeds the expected speed in still water, the current must be aiding the boat’s speed. To calculate it, we subtract the boat's speed in still water from the observed speed:
\[ v_{current} = v_{observed} - v_{still\,water} = 26.67 \, \text{m/s} - 20.0 \, \text{m/s} = 6.67 \, \text{m/s} \]
This formula breaks down easily:

• \(v_{observed}\): This is the speed measured in relation to the riverbank, at 26.67 m/s.
• \(v_{still\,water}\): This is the boat's speed in still water, at 20.0 m/s.
• \(v_{current}\): The result, 6.67 m/s, is the speed of the current, showing how much it boosts the boat's speed.

This simple subtraction provides an easy way to calculate the current’s speed, a vital component when navigating or analyzing a river's conditions.