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Chapter 3: Problem 34

Two piers, \(A\) and \(B,\) are located on a river: \(B\) is 1500 \(\mathrm{m}\) downstream from \(A\) (Fig. E3.34). 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 \(\mathrm{km} / \mathrm{h}\) relative to the water; the other walks on the shore at a constant speed of 4.00 \(\mathrm{km} / \mathrm{h}\) . The velocity of the river is 2.80 \(\mathrm{km} / \mathrm{h}\) in the direction from \(A\) to \(B .\) How much time does it take each person to make the round trip?

Short Answer

Expert verified
Walking takes 0.75 hours and rowing takes about 1.47 hours due to the river's current.

Step by step solution

01

Understand the Problem

We have two piers, A and B, 1500 meters apart. One person rows a boat and the other walks. Both travel at 4 km/h, but the river's current affects the boat. We need to find the round-trip time for each person.
02

Convert Units

The distance between the piers is 1500 meters, which is 1.5 kilometers. Speeds are given in km/h, so we ensure all distances and speeds are in these units.
03

Calculate Time for Walking

The walker covers a total of 3 kilometers (1.5 km to B and 1.5 km back to A) at a constant speed of 4 km/h. Time taken is given by the formula \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \). Thus, \( t_{\text{walk}} = \frac{3}{4} = 0.75 \) hours.
04

Calculate Net Speed for Boat

The rower has a rowing speed of 4 km/h relative to water. Downstream, the speed is boosted by the river's current to \( 4 + 2.8 = 6.8 \) km/h. Upstream, the speed is reduced by the current to \( 4 - 2.8 = 1.2 \) km/h.
05

Calculate Time for Rowing

The round trip consists of rowing 1.5 km downstream and 1.5 km upstream. The times are \( t_{\text{downstream}} = \frac{1.5}{6.8} \) and \( t_{\text{upstream}} = \frac{1.5}{1.2} \). The total time is the sum: \( t_{\text{row}} = \frac{1.5}{6.8} + \frac{1.5}{1.2} \).
06

Calculate Total Time for Rower

Calculating each segment's time, we have, \( t_{\text{downstream}} \approx 0.2206 \) hours and \( t_{\text{upstream}} \approx 1.25 \) hours. Thus, \( t_{\text{row}} \approx 0.2206 + 1.25 = 1.4706 \) hours.
07

Compare Both Times

The walker takes 0.75 hours for the round trip, while the rower takes approximately 1.47 hours in total for the same trip during both segments due to the influence of the current.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Kinematics
Kinematics is the branch of physics that studies motion without considering the forces that cause it. In this context, it describes how both the walker and the rower travel between piers A and B.
  • We analyze their motion by looking at distance and speed.
  • The time taken for each journey is calculated using the formula: \[\text{Time} = \frac{\text{Distance}}{\text{Speed}}\]
  • Units matter for consistent calculation; in this case, distance in kilometers and speed in kilometers per hour.
Kinematic calculations for the walker are straightforward as they travel at a constant speed and direction, making it a simple division. For the rower, the calculation becomes more complex due to varying velocities caused by the river current. This exercise serves as a great way to practice kinematic principles by applying them to real-world scenarios involving different modes of transportation.
River Current
The presence of a river current adds a layer of complexity to the rower's journey.
  • The current flows from A to B at 2.80 km/h, influencing the rowing speed.
  • When rowing downstream, the river aids the rower, increasing their effective velocity to 6.80 km/h (4.00 km/h + 2.80 km/h).
  • Conversely, rowing upstream is more challenging, reducing the effective velocity to 1.20 km/h (4.00 km/h - 2.80 km/h).
This means the time needed for the rower to complete the round trip is longer than simply going straight at 4.00 km/h, as they need to overcome the opposing flow while returning.
Understanding how the river current affects movement is crucial in kinematics, since real-life scenarios often involve factors that help or hinder motion, making it important to account for these when calculating travel times.
Round Trip Calculation
Calculating the time for a round trip involves adding the time taken for both legs of the journey, here from A to B and back to A.
  • For the walker, the trip is straightforward: 3 km at 4 km/h results in a simple calculation leading to 0.75 hours.
  • The rower's journey is divided into two: downstream and upstream.
  • Downstream, the time taken is calculated as \( t_{\text{downstream}} = \frac{1.5}{6.8} \), while upstream it's \( t_{\text{upstream}} = \frac{1.5}{1.2} \).
Adding these, the rower's total time is approximately 1.47 hours, showing the significant influence of the current.
This exercise outlines how round trip calculations are managed by considering each segment separately, then combining them. It also highlights the importance of external factors like the river current, illustrating how such elements affect the total time required. Understanding these calculations is essential for solving more complex problems in physics that involve multiple steps or varying conditions.

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