Problem 32
Question
Two piers, \(A\) and \(B\), are located on a river; \(B\) is 1500 m downstream from A (\(\textbf{Fig. E3.32}\)). Two friends must make round trips from pier \(A\) to pier \(B\) and return. One rows a boat at a constant speed of 4.00 km/h relative to the water; the other walks on the shore at a constant speed of 4.00 km/h. The velocity of the river is 2.80 km/h in the direction from \(A\) to \(B\). How much time does it take each person to make the round trip?
Step-by-Step Solution
Verified Answer
The rower takes 1.47 hours and the walker takes 0.75 hours to complete the round trip.
1Step 1: Convert the Distance to Kilometers
First, we need to convert the distance from meters to kilometers since the speeds are given in km/h. The distance between piers A and B is 1500 meters.Convert meters to kilometers: \( 1500 \text{ m} = 1.5 \text{ km} \)
2Step 2: Calculate the Rowing Time Downstream
When rowing downstream, the effective speed of the rower is the speed of the rower relative to the water plus the speed of the river. \[ \text{Effective downstream speed} = 4.00 \text{ km/h} + 2.80 \text{ km/h} = 6.80 \text{ km/h} \]The time taken to row between A and B downstream is: \[ \text{Time downstream} = \frac{1.5 \text{ km}}{6.80 \text{ km/h}} \approx 0.2206 \text{ h} \]
3Step 3: Calculate the Rowing Time Upstream
When rowing upstream, the effective speed of the rower is the speed of the rower relative to the water minus the speed of the river.\[ \text{Effective upstream speed} = 4.00 \text{ km/h} - 2.80 \text{ km/h} = 1.20 \text{ km/h} \]The time taken to row between B and A upstream is:\[ \text{Time upstream} = \frac{1.5 \text{ km}}{1.20 \text{ km/h}} = 1.25 \text{ h} \]
4Step 4: Calculate the Total Rowing Time
The total time for the rower to make the round trip is the sum of the time for the downstream and upstream travel.\[ \text{Total rowing time} = 0.2206 \text{ h} + 1.25 \text{ h} = 1.4706 \text{ h} \approx 1.47 \text{ h} \]
5Step 5: Calculate the Walking Time
The walker travels with a constant speed of 4.00 km/h, unaffected by the river's current.The total walking distance for the round trip is twice the distance between A and B.Round-trip walking distance: \(2 \times 1.5 \text{ km} = 3.0 \text{ km} \)The time it takes to walk the round trip is:\[ \text{Total walking time} = \frac{3.0 \text{ km}}{4.00 \text{ km/h}} = 0.75 \text{ h} \]
Key Concepts
Understanding Kinematics in Motion ProblemsInfluence of River Current on VelocityRound Trip Calculation for Different ModesConverting Distances for Consistent Units
Understanding Kinematics in Motion Problems
Kinematics is a branch of physics that deals with the motion of objects. It focuses on understanding the position, velocity, and acceleration of objects over time without considering the forces that cause the motion. In this particular exercise, we apply kinematics to analyze the motion of a rower and a walker moving between two points on a river. The core idea here is to distinguish between their speeds with respect to different frames of reference.
For instance, the rower's speed is relative to the water, which means it needs to be adjusted depending on the water flow. This requires understanding relative velocity, a key concept in kinematics: the velocity of an object with respect to another moving object. Similarly, the walker's motion is unaffected by the river, as it's straightforward on a solid ground.
The approach involves doing calculations such as finding effective speeds downstream and upstream by combining the velocity of the rower and the relative velocity of the river.
For instance, the rower's speed is relative to the water, which means it needs to be adjusted depending on the water flow. This requires understanding relative velocity, a key concept in kinematics: the velocity of an object with respect to another moving object. Similarly, the walker's motion is unaffected by the river, as it's straightforward on a solid ground.
The approach involves doing calculations such as finding effective speeds downstream and upstream by combining the velocity of the rower and the relative velocity of the river.
Influence of River Current on Velocity
River current can significantly affect the motion of objects moving through water. In our scenario, the current plays a pivotal role in the rower's journey. As the river has a current of 2.80 km/h going from pier A to pier B, it aids the boat's motion in that direction, effectively increasing the rowing speed.
Downstream motion is facilitated by the current, so you add the current's velocity to the rower’s velocity to get the effective speed. Conversely, rowing upstream is challenging because the current pushes against the vessel, which requires subtracting the current's speed from the rower's speed. This is critical because it affects not just the time taken but also the energy exerted when moving against or with the current.
Downstream motion is facilitated by the current, so you add the current's velocity to the rower’s velocity to get the effective speed. Conversely, rowing upstream is challenging because the current pushes against the vessel, which requires subtracting the current's speed from the rower's speed. This is critical because it affects not just the time taken but also the energy exerted when moving against or with the current.
- Downstream: Effective speed = Rower speed + River current
- Upstream: Effective speed = Rower speed - River current
Round Trip Calculation for Different Modes
Round trip calculations involve determining the total time taken for an individual to travel from a starting point to a destination and back. By examining both the rower and the walker, this problem highlights distinct dynamics.
The rower must account for both downstream and upstream times because river currents alter effective speeds. The downstream speed being faster than the upstream speed makes one leg of the journey quicker than the other. To find the total rowing time, you add the downstream and upstream times together.
In contrast, the walker experiences no such variance. Walking on land ensures a uniform speed throughout the journey since there are no external forces like a river current affecting it. Therefore, the walking round trip time calculation is straightforward as it merely involves the constant speed over double the distance.
The rower must account for both downstream and upstream times because river currents alter effective speeds. The downstream speed being faster than the upstream speed makes one leg of the journey quicker than the other. To find the total rowing time, you add the downstream and upstream times together.
In contrast, the walker experiences no such variance. Walking on land ensures a uniform speed throughout the journey since there are no external forces like a river current affecting it. Therefore, the walking round trip time calculation is straightforward as it merely involves the constant speed over double the distance.
Converting Distances for Consistent Units
Distance conversion is crucial for maintaining consistency in calculations, especially in physics problems where units may vary. The exercise begins with a distance given in meters but requires speed calculations in kilometers per hour. Thus, converting meters to kilometers ensures compatibility with the given velocity units.
This is achieved by understanding the basic metric conversion: 1 kilometer equals 1000 meters. Therefore, to convert meters to kilometers, you simply divide by 1000. For example, the distance from pier A to pier B is 1500 meters, which converts to 1.5 kilometers. This conversion is a fundamental step that must be done before solving the rest of the problem to avoid any discrepancies in unit mismatches.
Proper conversion consolidates the uniformity of units, which is critical for the accuracy and validity of any kinematic analysis or calculation.
This is achieved by understanding the basic metric conversion: 1 kilometer equals 1000 meters. Therefore, to convert meters to kilometers, you simply divide by 1000. For example, the distance from pier A to pier B is 1500 meters, which converts to 1.5 kilometers. This conversion is a fundamental step that must be done before solving the rest of the problem to avoid any discrepancies in unit mismatches.
Proper conversion consolidates the uniformity of units, which is critical for the accuracy and validity of any kinematic analysis or calculation.
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