1.80 You are hungry and decide to go to your favorite neighborhood fast-food restaurant. You leave your apartment and take the elevator 10 flights down (each flight is 3.0 m) and then go 15 m south to the apartment exit.

1.80 You are hungry and decide to go to your favorite neighborhood fast-food restaurant. You leave your apartment and take the elevator 10 flights down (each flight is 3.0 m) and then go 15 m south to the apartment exit. You then proceed 0.2 km east, turn north, and go 0.1 km to the entrance of the restaurant. (a) Determine the displacement from your apartment to the restaurant. Use unit vector notation for your answer, being sure to make clear your choice of coordinates. (b) How far did you travel along the path you took from your apartment to the restaurant, and what is the magnitude of the displacement you calculated in part (a)?

SOLUTION

a) Converting the $(10 flights )(\frac{3.0 \ m}{flight})=30.0 \ m$.

Let $\vec{A}=30.0 \ m$, down $= 30.0 \ m \ \hat{k}$

$\vec{B}=15 \ m$, south $=15 \ m \ \hat{i}$

$\vec{C}=0.2 \ km$, east $=200 \ m \ \hat{j}$

$\vec{D}=0.1 \ km$, east $=100 \ m \ (-\hat{i})$

then displacement, $\vec{d}=\vec{A}+\vec{B}+\vec{C}+\vec{D}$. Figure 1, shows a rough sketch of the situation.

Using unit vectors,

$\vec{d}=30.0 \ m \ \hat{k}+15 \ m \ \hat{i}+200 \ m \ \hat{j}-100 \ m \ \hat{i}$

$=-85 \ m \ \hat{i}+200 \ m \ \hat{j}+30.0 \ m \ \hat{k}$

b) Using the Pythagorean theorem to solve for the magnitude of vector $\vec{d}$,

$d=\sqrt{(-85 \ m)^2+(200 \ m)^2+(30.0 \ m)^2}=219 \ m \approx 220 \ m$

Since displacement is the shortest path traveled from the initial to the final position, the question "How far did you travel along the path you took from your apartment to the restaurant?" requires you to solve for total distance traveled from the initial to the final position regardless of direction. Since it's a scalar quantity, there's no arrow on top of the letter $d$,

$d = 30.0 \ m+ 15 \ m +200 \ m+100 \ m=345 \ m \approx 350 \ m$

As expected, total distance should be greater than displacement because you don't consider its direction.