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Chapter 2: Problem 10

In reaching her destination, a backpacker walks with an average velocity of \(1.34 \mathrm{m} / \mathrm{s},\) due west. This average velocity results because she hikes for \(6.44 \mathrm{km}\) with an average velocity of \(2.68 \mathrm{m} / \mathrm{s},\) due west, turns around, and hikes with an average velocity of \(0.447 \mathrm{m} / \mathrm{s},\) due east. How far east did she walk?

Short Answer

Expert verified
The backpacker walked approximately 3662 meters east.

Step by step solution

01

Define Given Information

Let's lay out the data we know: The average velocity \( V_{avg} = 1.34 \ \mathrm{m/s} \) due west, hiking distance west \( d_1 = 6.44 \ \mathrm{km} \) converted to meters \( 6440 \ \mathrm{m} \), average velocity west \( V_1 = 2.68 \ \mathrm{m/s} \), and average velocity east \( V_2 = 0.447 \ \mathrm{m/s} \).
02

Calculate Time of Travel West

For the westward travel, use the formula for time \( t = \frac{d}{V} \). Calculate \( t_1 \) as follows: \[ t_1 = \frac{6440 \ \mathrm{m}}{2.68 \ \mathrm{m/s}} \approx 2402.99 \ \mathrm{s} \]
03

Calculate Total Time of Journey

Utilize the average velocity to determine the total time: \[ t_{total} = \frac{d_{total}}{V_{avg}} \]. Because the entire journey averages at \( 1.34 \ \mathrm{m/s} \), with \( 6440 \ \mathrm{m} \) westward and some unknown eastward distance \( d_2 \), produce the equation \[ d_2 = t_{total} \times V_2 - t_1 \times V_1 \].
04

Express Total Time Equation

Write the total time as \( t_1 + t_2 = t_{total} \). Rearrange for eastward time \( t_2 = t_{total} - t_1 \). Use known quantities: \[ t_{total} = \frac{d_1 + d_2}{V_{avg}} = \frac{6440 \ \mathrm{m} + d_2}{1.34 \ \mathrm{m/s}} \].
05

Solve for Eastward Distance

Combine the equations \( t_2 = \frac{d_2}{0.447} \) and \( t_{total} = \frac{6440 \ \mathrm{m} + d_2}{1.34} \). Recollect \( t_2 = t_{total} - t_1 \): \[ \frac{d_2}{0.447} = \frac{6440 + d_2}{1.34} - 2402.99 \]. Solve for \( d_2 \approx 3662 \ \mathrm{m} \).

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Key Concepts

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

Distance Calculation
Calculating distance involves knowing either the speed and time of travel or deriving it using other parameters provided in a problem.
In this exercise, we know the backpacker walked west for 6.44 km, which we converted to 6440 meters. This is crucial for working with consistent units in calculations.
  • For the westward journey, the given distance is the starting point (6.44 km).
  • The eastward distance is unknown and is represented as \(d_2\).
To solve for the eastward distance, we rely on the average velocity formula and rearrange it to find the unknown eastward distance.
Time of Travel
Time of travel is calculated using the formula \(t = \frac{d}{V}\), where \(d\) is the distance and \(V\) is the velocity. In this problem, we determine the time taken to travel the known westward distance first.
  • The formula provided reveals how long the backpacker traveled west: \(t_1 = \frac{6440 \ \mathrm{m}}{2.68 \ \mathrm{m/s}} \approx 2402.99 \ \mathrm{s}\).
  • This calculation is straightforward since both the velocity and distance are known for the westward travel.
Next, total travel time is calculated using the average velocity over the entire journey, ensuring that every part of the trip is considered for an accurate assessment.
Velocity West and East
Understanding the role of velocity is key in determining how distances and time interact in this problem.
The backpacker's journey involves walking in two directions 鈥� west and east 鈥� each with different velocities.
  • When traveling west, the velocity is a constant 2.68 m/s over the 6440 meters.
  • Traveling east, a slower velocity of 0.447 m/s applies, representing a change in speed and direction.
The average velocity of the entire journey is given as 1.34 m/s west, offering a crucial piece of data in solving for both time and distance traveled eastward.
Problem-Solving Steps
Approaching this problem involves a structured series of equations and logical reasoning:
  • Start by determining all given data and understand the direction of travel for each segment.
  • Calculate the travel time available using the average velocity for the entire trip and known distances.
  • Express the total travel time in terms of westward and eastward travel: \(t_{total} = t_1 + t_2\).
  • Solve equations to express the unknown eastward distance \(d_2\) using the established formula: \(\frac{d_2}{0.447} = \frac{6440 + d_2}{1.34} - 2402.99\).
  • By solving, find that the distance eastward is approximately 3662 meters.
Through these steps, solving for an unknown component of motion becomes logical and achievable, emphasizing the importance of carefully using all provided average and directional velocities with the correct formulas.

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