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Chapter 3: Problem 38

In its daily prowl of the neighborhood, a cat makes a displacement of \(120 \mathrm{m}\) due north, followed by a \(72-\mathrm{m}\) displacement due west. (a) Find the magnitude and direction of the displacement required for the cat to return home. (b) If, instead, the cat had first prowled \(72 \mathrm{m}\) west and then \(120 \mathrm{m}\) north, how would this affect the displacement needed to bring it home? Explain.

Short Answer

Expert verified
The return displacement is approximately 139.85 m at 59.04掳 south of east. Changing the order of movements does not affect the displacement required to return home.

Step by step solution

01

Understand the Problem

We are tasked with finding the displacement required for a cat to return to its starting point after moving 120 m north and 72 m west. We must calculate the magnitude and direction of this return displacement.
02

Visualize the Path as a Vector Triangle

Imagine the northward and westward displacements as two sides of a right triangle. The straight path back to the starting point is the hypotenuse of this triangle. The northward displacement is the vertical leg (120 m), and the westward displacement is the horizontal leg (72 m).
03

Calculate the Magnitude of the Return Displacement

Apply the Pythagorean theorem to find the hypotenuse (magnitude of the return displacement): \[ c = \sqrt{(120)^2 + (72)^2} = \sqrt{14400 + 5184} = \sqrt{19584} \approx 139.85 \, \text{m} \] Thus, the magnitude of the displacement required for the cat to return is approximately 139.85 m.
04

Determine the Direction Using Trigonometry

Use trigonometry to find the angle of the return path with respect to the west direction. Calculate the angle \( \theta \) with tangent:\[ \tan \theta = \frac{120}{72} \] \[ \theta = \tan^{-1}\left(\frac{120}{72}\right) \approx 59.04^\circ \] The direction is 59.04掳 south of east (in the southwest quadrant), or equivalently, 30.96掳 west of south.
05

Analyze the Effect of Changing the Order of Movement

If the cat first moves 72 m west then 120 m north, the return displacement remains the same because displacement is independent of path. The resultant vector's magnitude and direction do not change with different movement sequences due to the commutative property of vector addition.

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Key Concepts

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

Pythagorean Theorem
The Pythagorean theorem is a fundamental concept in mathematics that relates the sides of a right triangle. If you know the lengths of two sides of a right triangle, you can find the length of the third side using this theorem. It鈥檚 expressed as follows:
  • Given a right triangle with legs of lengths \( a \) and \( b \), and hypotenuse \( c \), the equation is \( c^2 = a^2 + b^2 \).
  • In our problem, the cat's movement north and west represents the two legs of the triangle.
  • We calculated the hypotenuse to determine the straight-line path back to the starting point.
Applying the formula helps us find the shortest distance back, called the magnitude of displacement, regardless of the path taken. The theorem is a crucial step for solving problems involving right triangles and analyzing vector movement.
Trigonometry
Trigonometry deals with the relationships between angles and sides of triangles. It is especially useful for calculating angles that are not so easy to measure directly. In this exercise, we used it to find the direction of the displacement vector necessary for the cat's return.
  • Trigonometry functions like tangent, sine, and cosine help in relating side lengths to angles.
  • For our problem, we used the tangent function: \( \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \).
  • Here, the opposite side is the cat's northern walk (120 m), and the adjacent side is the westward walk (72 m).
By computing \( \theta = \tan^{-1}\left(\frac{120}{72}\right) \), we determine the angle for the cat's return journey, giving us a clear direction to guide our understanding of the problem. Trigonometry simplifies finding direction, turning angles and distances into comprehensive visual data.
Vector Addition
Vector addition is the process of combining two or more vectors to determine a resultant vector. It allows one to understand the net effect of multiple movements or forces. In the problem, the cat's northward and westward displacements can be thought of as vectors.
  • By adding these vectors, the resultant gives us the direct line back home.
  • This resultant vector, calculated using both the Pythagorean theorem for magnitude and trigonometry for direction, shows that path order doesn鈥檛 change the final vector.
  • The commutative property means the sum remains the same, no matter whether we go north first and then west, or west and then north.
Understanding vector addition can dramatically simplify many physics and engineering problems. By visualizing movements as vectors, solutions become more apparent and manageable.

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Most popular questions from this chapter

You drive a car 1500 ft to the east, then \(2500 \mathrm{ft}\) to the north. If the trip took 3.0 minutes, what were the direction and magnitude of your average velocity?

A vector \(\overrightarrow{\mathrm{A}}\) has a magnitude of \(40.0 \mathrm{m}\) and points in a direction \(20.0^{\circ}\) below the positive \(x\) axis. A second vector, \(\overrightarrow{\mathbf{B}},\) has a magnitude of \(75.0 \mathrm{m}\) and points in a direction \(50.0^{\circ}\) above the positive \(x\) axis. (a) Sketch the vectors \(\overrightarrow{\mathbf{A}}, \overrightarrow{\mathbf{B}},\) and \(\overrightarrow{\mathrm{C}}=\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}\). (b) Using the component method of vector addition, find the magnitude and direction of the vector \(\overrightarrow{\mathrm{C}}\).

The pilot of an airplane wishes to fly due north, but there is a \(65-\mathrm{km} / \mathrm{h}\) wind blowing toward the east. (a) In what direction should the pilot head her plane if its speed relative to the air is \(340 \mathrm{km} / \mathrm{h} ?\) (b) Draw a vector diagram that illustrates your result in part (a). (c) If the pilot decreases the air speed of the plane, but still wants to head due north, should the angle found in part (a) be increased or decreased?

An air traffic controller observes two airplanes approaching the airport. The displacement from the control tower to plane 1 is given by the vector \(\bar{A},\) which has a magnitude of \(220 \mathrm{km}\) and points in a direction \(32^{\circ}\) north of west. The displacement from the control tower to plane 2 is given by the vector \(\overrightarrow{\mathbf{B}}\), which has a magnitude of \(140 \mathrm{km}\) and points \(65^{\circ}\) east of north. (a) Sketch the vectors \(\overrightarrow{\mathbf{A}},-\overrightarrow{\mathbf{B}},\) and \(\overrightarrow{\mathbf{D}}=\overrightarrow{\mathbf{A}}-\overrightarrow{\mathbf{B}} .\) Notice that \(\overrightarrow{\mathbf{D}}\) is the displacement from plane 2 to plane \(1 .\) (b) Find the magnitude and direction of the vector \(\overrightarrow{\mathbf{D}}\).

As you hurry to catch your flight at the local airport, you encounter a moving walkway that is \(85 \mathrm{m}\) long and has a speed of \(2.2 \mathrm{m} / \mathrm{s}\) relative to the ground. If it takes you \(68 \mathrm{s}\) to cover \(85 \mathrm{m}\) when walking on the ground, how long will it take you to cover the same distance on the walkway? Assume that you walk with the same speed on the walkway as you do on the ground.
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