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Chapter 1: Problem 52

Two geological field teams are working in a remote area. A global positioning system (GPS) tracker at their base camp shows the location of the first team as \(38 \mathrm{km}\) away, \(19^{\circ}\) north of west, and the second team as 29 km away, \(35^{\circ}\) east of north. When the first team uses its GPS to check the position of the second team, what does the GPS give for the second team's (a) distance from them and (b) direction, measured from due east?

Short Answer

Expert verified
(a) 53.70 km, (b) 12.2掳 north of east.

Step by step solution

01

Break Down the Problem

First, identify what is being asked. We need to calculate "(a) the distance between the two groups" and "(b) the direction of the second team from the first team, as measured from due east." We will do this by calculating the positions as vectors, and then finding their difference.
02

Translate Positions into Vectors

The positions of the teams can be expressed using vectors. For the first team's position: \[\begin{align*}\text{West Component} &= 38 \cos 19^{\circ},\\text{North Component} &= 38 \sin 19^{\circ}.\end{align*}\]For the second team: \[\begin{align*}\text{East Component} &= 29 \cos 35^{\circ},\\text{North Component} &= 29 \sin 35^{\circ}.\end{align*}\]
03

Express Team Locations as Coordinates

Calculate the cartesian coordinates for each team relative to the base camp.Team 1: West is negative on the x-axis, so: \[(x_1, y_1) = (-38 \cos 19^{\circ}, 38 \sin 19^{\circ})\]Team 2:East is positive on the x-axis, so: \[(x_2, y_2) = (29 \sin 35^{\circ}, 29 \cos 35^{\circ})\]
04

Distance Calculation Using Vector Difference

Calculate the vector difference between the two teams to determine the displacement vector.\[\mathbf{d} = (x_2-x_1, y_2-y_1)\]The distance, \(D\), between them is the magnitude of \(\mathbf{d}\):\[D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]
05

Calculate the Direction of the Second Team

Find the angle \(\theta\) by calculating the angle of \(\mathbf{d}\) relative to the positive x-axis (East).\[\theta = \tan^{-1}\left(\frac{y_2-y_1}{x_2-x_1}\right)\]This angle gives the direction of the second team measured from due east.
06

Solve the Equations

Substitute all known values into the equations:For Team 1: \[(x_1, y_1) = (-38 \cos 19^{\circ}, 38 \sin 19^{\circ})\]\[(x_1, y_1) = (-35.88, 12.38)\]For Team 2: \[(x_2, y_2) = (29 \sin 35^{\circ}, 29 \cos 35^{\circ})\]\[(x_2, y_2) = (16.63, 23.77)\]Calculate displacement: \[(x_2-x_1, y_2-y_1) = (52.51, 11.39)\]
07

Calculate Distance and Direction

Find the distance using the magnitude of the displacement vector:\[D = \sqrt{52.51^2 + 11.39^2} = 53.70 \text{ km}\]Find the direction:\[\theta = \tan^{-1}\left(\frac{11.39}{52.51}\right) \approx 12.2^{\circ} \text{ north of east}\]

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Key Concepts

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

Angles and Directions
Understanding angles and directions is central to solving problems involving vectors, like tracking the locations of teams in a field. To comprehend this, imagine angles as a tool to tell you where to point, while direction is what guides you along that path.

In this exercise, each team's position is described by an angle coupled with a cardinal direction (e.g., north, east, west). This combination provides a precise navigation instruction. For instance, "19掳 north of west" directs you slightly towards north while primarily moving west. This means you start heading due west and adjust 19掳 towards the north.

Angles are measured degrees, helping specify direction relative to a reference direction, often the cardinal directions. When working with vector problems, drawing a diagram can help visualize where the angle takes you. Visual aids are especially helpful in translating these directional angles into useful information, letting you understand the vector's path.
Vectors in Physics
Vectors are crucial in physics for representing quantities with both magnitude and direction, like velocity and force. In our exercise, we鈥檙e dealing with position vectors that describe teams' locations in the field.

Each team鈥檚 location is expressed as a vector. The vector has two components: one in the east-west direction, and another in the north-south direction.
  • The components can be found using trigonometry. For example, if you know the distance and angle, such as 38 km at 19掳 north of west, you can use trigonometric functions (cosine and sine) to find the exact horizontal and vertical components of this vector.

It鈥檚 like breaking down the vector into smaller pieces that add up to give you the direction and distance the vector represents. This decomposition into components allows for easier manipulation and calculation, such as adding vectors to find resultant positions.
Displacement and Distance
Displacement and distance might sound similar but mean different things in physics. Distance refers to how much ground is covered, without worrying about direction. Displacement, however, measures the change in position in a specific direction.

For the field teams, displacement is the vector difference between their positions, telling us not just the distance between them, but also the direction from one to another. The vector difference is calculated using the components found earlier.
  • Displacement is represented as a vector, giving both the magnitude (how far) and direction (which way).
  • The actual distance they need to cover to meet each other is the length of the displacement vector, calculated using the Pythagorean theorem.

Understanding both concepts is essential, especially in problems where direction changes or isn鈥檛 straight, helping illustrate the efficiency or effect of paths taken by teams. Comprehending these terms helps in a variety of applications beyond just determining positions.

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