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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
She walked 3.60 km east.

Step by step solution

01

Understand the Problem

The backpacker walks with an average velocity of \(1.34 \mathrm{m/s}\) towards the west. She initially walks westward for \(6.44 \mathrm{km}\) at \(2.68 \mathrm{m/s}\). Then, she turns and walks east at \(0.447 \mathrm{m/s}\). We need to determine the distance she walked eastward.
02

Convert Kilometers to Meters

Convert the distance hiked westward into meters since the given velocities are in meters per second. \[6.44 \text{ km} = 6440 \text{ m}\]
03

Solve for Time Hiking Westward

Calculate the time taken to hike \(6.44 \mathrm{km}\) west using the formula: \[\text{Time} = \frac{\text{Distance}}{\text{Velocity}}\] \[\text{Time} = \frac{6440 \text{ m}}{2.68 \text{ m/s}} \approx 2402.99 \text{ s}\]
04

Calculate Total Time of the Journey

Use the average velocity formula: \[\text{Total Time} = \frac{\text{Total Displacement}}{\text{Average Velocity}}\]Since direction is important here, treat west as positive and east as negative. Let \(d\) be the eastward distance. \[\frac{6440 - d}{1.34} = 2402.99 + \frac{d}{0.447}\]
05

Set Up the Equation

Rearrange and solve for \(d\).\[\frac{6440 - d}{1.34} = 2402.99 + \frac{d}{0.447} \] Simplify the equation to:\[\frac{6440}{1.34} - \frac{d}{1.34} = 2402.99 + \frac{d}{0.447}\]
06

Solve the Equation

Bring all \(d\) terms to one side. Calculate:\[\frac{6440}{1.34} - 2402.99 = \frac{d}{1.34} + \frac{d}{0.447} \] Solve for \(d\):\[4809.7 - 2402.99 = \frac{d(0.447 + 1.34)}{0.447 \times 1.34}\] \[d \approx 3602.52 \text{ m} \]
07

Convert to Kilometers (Optional)

Since the problem might require an answer in kilometers, convert meters back to kilometers:\[d \approx 3.60 \text{ km} \]

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Key Concepts

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

Distance Calculation
Distance calculation is a fundamental concept that tells us how much ground an object has covered during its motion. In this exercise, the backpacker walks west and then east. To determine the total distance traveled, we need to consider the total length of each leg of the trip, irrespective of direction.

When calculating distance:
  • The distance covered during the westward hike is already given in the problem: it's 6.44 kilometers. This measurement shows how far the backpacker initially moved west.
  • Next, the distance for the eastward part of the hike needed to be computed through solving a mathematical equation. This distance ended up being approximately 3.60 kilometers east.
In the end, the total distance includes both the journey west and the journey east. The concept of distance focuses purely on how much path the backpacker took physically, regardless of the starting and ending points.
Conversion of Units
Converting units is essential when dealing with measurements in different units. In physics problems, units need to be consistent to ensure calculations are precise and meaningful. In this exercise, we're dealing with distance in kilometers and velocities in meters per second. It's easier to perform calculations when distances match the velocities in meters.

To convert kilometers to meters:
  • Remember that 1 kilometer is equal to 1000 meters. Thus, the backpacker's westward hike of 6.44 kilometers is converted into 6440 meters. This conversion keeps units consistent and simplifies calculation with velocities given in meters per second.
Unit conversion is a straightforward yet crucial step in problem-solving, ensuring that all your numbers speak the same language before embarking on further calculations.
Displacement
Displacement is quite different from distance. While distance measures the total path covered, displacement focuses on the change in position from the starting point to the final point. It considers the shortest path between these two points and is directional, meaning it has both magnitude and direction.

In this context:
  • The backpacker's displacement is calculated by considering the initial and final positions relative to her starting point.
  • Because the backpacker hikes in two directions, it's important to treat the westward direction as positive and the eastward direction as negative.
  • The total displacement equation \[ \frac{6440 - d}{1.34} = 2402.99 + \frac{d}{0.447} \] is solved to find the change in position due to her eastbound hike. Solving this equation helps us find out exactly how far she displaced herself from her original westward travel.
Understanding displacement is fundamental as it helps in understanding motion in terms of changing positions, rather than just the total ground covered.

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Most popular questions from this chapter

An astronaut on a distant planet wants to determine its acceleration due to gravity. The astronaut throws a rock straight up with a velocity of \(+15 \mathrm{m} / \mathrm{s}\) and measures a time of \(20.0 \mathrm{s}\) before the rock returns to his hand. What is the acceleration (magnitude and direction) due to gravity on this planet?

A cheetah is hunting. Its prey runs for \(3.0 \mathrm{s}\) at a constant velocity of \(+9.0 \mathrm{m} / \mathrm{s} .\) Starting from rest, what constant acceleration must the cheetah maintain in order to run the same distance as its prey runs in the same time?

The leader of a bicycle race is traveling with a constant velocity of \(+11.10 \mathrm{m} / \mathrm{s}\) and is \(10.0 \mathrm{m}\) ahead of the second-place cyclist. The secondplace cyclist has a velocity of \(+9.50 \mathrm{m} / \mathrm{s}\) and an acceleration of \(+1.20 \mathrm{m} / \mathrm{s}^{2}\). How much time elapses before he catches the leader?

A tourist being chased by an angry bear is running in a straight line toward his car at a speed of \(4.0 \mathrm{m} / \mathrm{s}\). The car is a distance \(d\) away. The bear is \(26 \mathrm{m}\) behind the tourist and running at \(6.0 \mathrm{m} / \mathrm{s}\). The tourist reaches the car safely. What is the maximum possible value for \(d\) ?

A race driver has made a pit stop to refuel. After refueling, he starts from rest and leaves the pit area with an acceleration whose magnitude is \(6.0 \mathrm{m} / \mathrm{s}^{2}\), after \(4.0 \mathrm{s}\) he enters the main speedway. At the same instant, another car on the speedway and traveling at a constant velocity of \(70.0 \mathrm{m} / \mathrm{s}\) overtakes and passes the entering car. The entering car maintains its acceleration. How much time is required for the entering car to catch the other car?
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