Q72P - University Physics with Modern Physics

Question

You decide to go to your favourite neighbourhood restaurant. You leave your apartment, take the elevator 10 flights down (each flight is 3.0 m), and then walk 15 m south to the apartment exit. You then proceed 0.200 km east, turn north, and walk 0.100 km to the entrance of the restaurant. (a) Determine the displacement from your apartment to the restaurant. Use unit vector notation for your answer, clearly indicating 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)?

Step-by-Step Solution

Answer

a)The displacement from apartment to the restaurant is,$\left(200 \mathrm{m}\right)\hat{\mathrm{i}}+\left(85\mathrm{m}\right)\hat{\mathrm{j}}-\left(30\mathrm{m}\right)\hat{\mathrm{k}}$

b)The magnitude of the resultant of the total displacement is,$219\mathrm{m}$ and the I have travelled is 345m.

Step 1: Identification of given data

The given data can be listed below as-

• The vector A is expressed as, $\overrightarrow{A}\left(10flights\right)\left(\frac{3m}{1flight}\right)\left(-\hat{k}\right)=-\left(30m\right)\hat{k}$
• The vector B is expressed as, $\overrightarrow{\mathrm{B}}=-\left(15\mathrm{M}\right) \hat{\mathrm{j}}$
• The vector C is expressed as, $\overrightarrow{\mathrm{C}}=\left(0.2\mathrm{km}\right)\left(\frac{1000\mathrm{m}}{1\mathrm{km}}\right)\left(\hat{\mathrm{i}}\right)=\left(200\mathrm{m}\right) \hat{\mathrm{i}}$
• The vector D is expressed as, $\overrightarrow{\mathrm{D}}=\left(0.1\mathrm{km}\right)\left(\frac{1000\mathrm{m}}{1\mathrm{km}}\right)\left(\hat{\mathrm{j}}\right)=\left(100\mathrm{m}\right) \hat{\mathrm{j}}$

Step 2: Concept of resultant displacement

A resultant displacement is the outcome of adding two or more displacement vectors together. 

Any two vectors, however, may be combined if their vector quantities are equal. A resultant velocity is formed when two or even more velocity vector is combined. A resultant force is the sum of two or maybe more force vectors.

Step 3: Determination of the displacements

a)

The equation of the four displacement vector sum can be evaluated by,

$\overrightarrow{R}=\overrightarrow{A}+\overrightarrow{B}+\overrightarrow{C}+\overrightarrow{D}$ 

Here,$\overrightarrow{R}$ is the resultant vector

 

Substituting the values in the above equation,

 $$\begin{gathered} \overrightarrow{\mathrm{R}}=\left(-30\mathrm{m}\right)\hat{\mathrm{k}}+\left(-15\mathrm{m}\right)\hat{\mathrm{j}}+\left(200\mathrm{m}\right)\hat{\mathrm{i}}+\left(100\mathrm{m}\right)\hat{\mathrm{j}} \\ =\left(200\mathrm{m}\right)\hat{\mathrm{i}}+\left(85\mathrm{m}\right)\hat{\mathrm{j}}-\left(30\mathrm{m}\right)\hat{\mathrm{k}} \end{gathered}$$

 Thus, the sum of the four vector is,$\left(200m\right)\hat{i}+\left(85m\right)\hat{j}-\left(30m\right)\hat{k}$

Step 4: Determine the magnitude of resultant displacement

b)

 The magnitude of resultant displacement can be evaluated by,

The equation of the sum of the distances of the individual segments is,

$\mathrm{S}=\mathrm{A}+\mathrm{B}+\mathrm{C}+\mathrm{D}$

 Here, S is the sum of the distances

 Substituting the values in the above equation,

 $30\mathrm{m}+15\mathrm{m}+200\mathrm{m}+100\mathrm{m}=345\mathrm{m}$

 The equation of the magnitude of the resultant of the total displacement is,

$\mathrm{R}=\sqrt{{\mathrm{R}}_{\mathrm{x}}^2+{\mathrm{R}}_{\mathrm{y}}^2+{\mathrm{R}}_{\mathrm{z}}^2}$

 Here, R is the magnitude of the resultant 

 

Substituting the values in the above equation,

$$\begin{gathered} \mathrm{R}=\sqrt{{\left(200\mathrm{m}\right)}^2{\left(85\mathrm{m}\right)}^2+{\left(-30\mathrm{m}\right)}^2} \\ = 219\mathrm{m} \end{gathered}$$ 

 Thus, the magnitude of the resultant of the total displacement is, $219\mathrm{m}$and I have travelled along the path that i took from my apartment to the restaurant was $345\mathrm{m}$.