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College Physics (Urone)
Found in: Page 121
College Physics (Urone)

College Physics (Urone)

Book edition 1st Edition
Author(s) Paul Peter Urone
Pages 1272 pages
ISBN 9781938168000

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Short Answer

Suppose you first walk 12.0 m in a direction 20º west of north and then 20.0 m in a direction 40.0º south of west. How far are you from your starting point and what is the compass direction of a line connecting your starting point to your final position? (If you represent the two legs of the walk as vector displacements A and B , as in Figure 3.56, then this problem finds their sum R = A + B.)

You are at \(19.5\;{\rm{m}}\) from the starting point and direction of compass is \(4.65^\circ \) from west of south.

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: Identification of given data

The displacement along west of north is \(A = 12\;{\rm{m}}\)

The direction of displacement along west of north is\(\alpha = 20^\circ \).

The displacement along south of west is \(B = 20\;{\rm{m}}\)

The direction of displacement along south of west is\(\beta = 40^\circ \).

The displacement is resolved along horizontal and vertical direction then sum of all the horizontal components gives horizontal component of net displacement. The sum of vertical components gives vertical component of net displacement.

Step 2: Determination of distance from starting point

The horizontal component of the net displacement is given as:

\(\begin{align}{}{R_x} &= - A\cos \left( {90^\circ - \alpha } \right) - B\cos \beta \\{R_x} &= - \left( {A\sin \alpha + B\cos \beta } \right)\end{align}\)

Substitute all the values in the above equation.

\(\begin{align}{}{R_x} &= - \left( {\left( {12\;{\rm{m}}} \right)\left( {\sin 20^\circ } \right) + \left( {20\;{\rm{m}}} \right)\left( {\cos 40^\circ } \right)} \right)\\{R_x} &= - 19.43\;{\rm{m}}\end{align}\)

The vertical component of the net displacement is given as:

\(\begin{align}{}{R_y} &= A\sin \left( {90^\circ - \alpha } \right) - B\sin \beta \\{R_y} &= A\cos \alpha - B\sin \beta \end{align}\)

Substitute all the values in the above equation.

\(\begin{align}{}{R_y} &= \left( {12\;{\rm{m}}} \right)\left( {\cos 20^\circ } \right) - \left( {20\;{\rm{m}}} \right)\left( {\sin 40^\circ } \right)\\{R_y} &= - 1.58\;{\rm{m}}\end{align}\)

The distance from starting point is given as:

\(R = \sqrt {R_x^2 + R_y^2} \)

Substitute all the values in the above equation.

\(\begin{align}{}R &= \sqrt {{{\left( { - 19.43\;{\rm{m}}} \right)}^2} + {{\left( { - 1.58\;{\rm{m}}} \right)}^2}} \\R &= 19.5\;{\rm{m}}\end{align}\)

Step 3: Determination of direction from starting point

The direction of compass from starting point is given as:

\(\tan \theta = \frac{{{R_y}}}{{{R_x}}}\)

Substitute all the values in the above equation.

\(\begin{align}{}\tan \theta = \frac{{ - 1.58\;{\rm{m}}}}{{ - 19.43\;{\rm{m}}}}\\\theta = 4.65^\circ \end{align}\)

Therefore, you are at \(19.5\;{\rm{m}}\) from the starting point and direction of compass is \(4.65^\circ \) from west of south.

Most popular questions for Physics Textbooks

Repeat the problem above, but reverse the order of the two legs of the walk; show that you get the same final result. That is, you first walk leg \({\rm{B}}\), which is \(20.0\;{\rm{m}}\) in a direction exactly \(40^\circ \) south of west, and then leg \({\rm{A}}\), which is \(12.0\;{\rm{m}}\) in a direction exactly \(20^\circ \) west of north. (This problem shows that \({\rm{A}} + {\rm{B}} = {\rm{B}} + {\rm{A}}\).)

A farmer wants to fence off his four-sided plot of flat land. He measures the first three sides, shown asA ,B , andC in Figure 3.62, and then correctly calculates the length and orientation of the fourth side D. What is his result?

Find the magnitudes of velocities \({v_A}\) and \({v_B}\) in Figure 3.57.

The two velocities \({{\rm{v}}_{\rm{A}}}\) and \({{\rm{v}}_{\rm{B}}}\) add to give a total \({{\rm{v}}_{{\rm{tot}}}}\) .

A new landowner has a triangular piece of flat land she wishes to fence. Starting at the west corner, she measures the first side to be \(80.0{\rm{ m}}\) long and the next to be \(105\;{\rm{m}}\). These sides are represented as displacement vectors \({\rm{A}}\) from \({\rm{B}}\) in Figure 3.61. She then correctly calculates the length and orientation of the third side \({\rm{C}}\). What is her result?

Find the following for path C in Figure

(a) the total distance traveled and

(b) the magnitude and direction of the displacement from start to finish.

In this part of the problem, explicitly show how you follow the steps of the analytical method of vector addition.

Figure: The various lines represent paths taken by different people walking in a city. All blocks are \(120{\rm{ m}}\) on a side.
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