Q32E - University Physics with Modern Physics

Question

A postal employee drives a delivery truck over the route shown in Fig.below. Use the method of components to determine the magnitude and direction of her resultant displacement. In a vector-addition diagram (roughly to scale), show that the resultant displacement found from your diagram is in qualitative agreement with the result you obtained by using the method of components.

Step-by-Step Solution

Answer

As per mentioned all three components, Resultant and its magnitude, direction is calculated accordingly

Step1: Explaination of Diagram

As showing above fig-(measurement and angle )it is asked that the resultant displacement found from diagram.

There are three displacement figure A,B and C,assume that the y axis points to the north and x axis points to east.

Step 2: Calculation component of A, B and C

$$\begin{gathered} \cos 45°=\frac{cx}{3.1} \\ Cx=3.1 \cos 45°\approx 2.19 \\ \sin 45°=\frac{Cy}{3.1} \\ Cy=3.1Sin 45°\approx 2.19 \end{gathered}$$

Vectors are

$$\begin{gathered} A=\langle Ax,Ay\rangle =\langle 0,2.6\rangle km \\ B=\langle Bx,By\rangle =\langle 4.0,0\rangle km \\ C=\langle Cx,Cy\rangle =\langle 3.1Cos45°,3.1Sin 45°\rangle km \end{gathered}$$

Step 3: Calculation of Resultant Vector

R = A+B+C

R =$(0,2.6)+(4.0,0)+3.1Cos45,3.1Sin 45)$

R = $(0+4.0+3.1Cos45 ,2.6+0+3.1Sin 45)$

R$\approx (6.19, 4.79)km$……………………….(1)

Step 4:Calculation of Magnitude of Resultant

$\left|R\right|=\sqrt{(6.19{)}^2+(4.79{)}^2}$km ………….from equation (1)

$\mathbf{\approx }\mathbf{7}\mathbf{.}\mathbf{83}\mathit{k}\mathit{m}$

Step 5:Calculation of direction

$$\begin{gathered} \theta =ta{n}^{-1}\frac{(2.6+3.1Sin45)}{(4.0+3.1Cos45)} \\ \approx ta{n}^{-1}\frac{(4.79)}{(6.19)} \\ \mathit{\theta }\mathbf{\approx }\mathbf{37}\mathbf{.}\mathbf{7}\mathbf{°} \end{gathered}$$

Step5: Conclusion

As per mentioned all three components, Resultant and its magnitude, direction is calculated accordingly