Question

7. Draw each of the following vectors. Then find its x - and y- components.

a. $\overrightarrow{v}=\left(7.5\text{m}/\text{s},{30}^{°}\right.$clockwise from the positive x-axis )

b. $\overrightarrow{a}=\left(1.5\text{m}/{\text{s}}^2,{30}^{°}\right.$above the negative x-axis )

c. $\overrightarrow{F}=\left(50.0\text{N},{36.9}^{°}\right.$counter clockwise from the positive y-axis )

Short answer

a. x-component $=3.75\text{m}/\text{s}$

y-component$=6.5\text{m}/\text{s}$

b. x-component$=-1.3\text{m}/{\text{s}}^2$

y-component$=0.75\text{m}/{\text{s}}^2$$=0.75\text{m}/{\text{s}}^2$

C. x-component$=-30\text{N}$

y-component$=40\text{N}$

Step-by-step solution

Step 1.  Given information

a.The velocity vector is $\overrightarrow{v}=\left(7.5\text{m}/\text{s},{30}^{°}\right.$clockwise from the positive y-axis )

b. The acceleration vector is$\overrightarrow{a}=\left(1.5\text{m}/{\text{s}}^2,{30}^{°}\right.$ above the negative -axis)

c. The force vector is data-custom-editor="chemistry" $\overrightarrow{F}=\left(50\text{N},{36.9}^{°}\right.$counterclockwise from the positive y-axis )

v→Part (a)

Vector can be represented as below:

x- component is

$\begin{array}{l}{v}_{\text{x}}=v\sin \theta \\ =7.5\text{m}/\text{s}\times \sin {30}^{°}\\ =7.5\text{m}/\text{s}\times 0.5\\ =3.75\text{m}/\text{s}\end{array}$

y-component is

$\begin{array}{l}{v}_{\text{y}}=v\cos \theta =7.5\text{m}/\text{s}\times \cos {30}^{°}\\ =7.5\text{m}/\text{s}\times 0.866\\ =6.495\text{m}/\text{s}\\ \approx 6.5\text{m}/\text{s}\end{array}$

Part (b)

Vector $\overrightarrow{a}$can be represented as below:

x-component is

$\begin{array}{l}{a}_x=-a\cos \theta =-1.5\text{m}/{\text{s}}^2\times \cos {30}^{°}\\ =-1.5\text{m}/{\text{s}}^2\times 0.866\\ =-1.299\text{m}/{\text{s}}^2\\ =-1.3\text{m}/{\text{s}}^2\end{array}$

y-component is

$\begin{array}{l}{a}_y=a\sin \theta \\ =1.5\text{m}/{\text{s}}^2\times \sin {30}^{°}\\ =1.5\text{m}/{\text{s}}^2\times 0.5\\ =0.75\text{m}/{\text{s}}^2\end{array}$

Part(c)

Vector $\overrightarrow{F}$can be represented as below:

x-component is

$\begin{array}{l}{F}_x=-F\sin \theta \\ =-50\text{N}\times \sin {36.9}^{°}\\ =-50\text{N}\times 0.60\\ =-30\text{N}\end{array}$

y-component is

$\begin{array}{l}{F}_y=F\cos \theta \\ =50\text{N}\times \cos {36.9}^{°}\\ =50\text{N}\times 0.799\\ =39.98\text{N}\\ \approx 40\text{N}\end{array}$