Vector Addition and Subtraction| Analytical Method| College Physics| Problem 3.16

Solve the following problem using analytical techniques: Suppose you walk 18.0 m straight west and then 25.0 m straight north. 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 3.60, then this problem asks you to find their sum R = A + B .)

3.60
Figure3.60 The two displacements A and B add to give a total displacement R having magnitude R and direction θ.

Note that you can also solve this graphically. Discuss why the analytical technique for solving this problem is potentially more accurate than the graphical technique.

SOLUTION:

Consider the right triangle formed by the legs A, B, and R. We know that A is 18 m, B is 25 m, and we are solving for the magnitude of R. We can do this by using the Pythagorean Theorem. That is

R=\sqrt{A^2+B^2}

R=\sqrt{\left(18\:m\right)^2+\left(25\:m\right)^2}

R=\sqrt{324+625}

R=\sqrt{949}

R=30.8\:m

So, the distance is about 30.8 meters from the starting point. To solve for the value of the unknown angle, θ, we can use the tangent function. That is 

tan\:\theta =\frac{B}{A}

tan\:\theta =\frac{25\:m}{18\:m}

\theta =tan^{-1}\left(\frac{25}{18}\right)

\theta =54.25^{\circ}

So, the compass reading, can be solved by taking the complimentary angle, \phi as shown in the figure.

  3.4

\phi =90^{\circ} -54.25^{\circ}

\phi =35.75^{\circ}

Therefore, the compass angle is 35.75^{\circ} \:West\:of\:North

 

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