Question

Vectors and Complex numbers Part AYou drive 30 miles due east in a half hour. Then, you turn left and drive 30 miles north in 1 hour. What are your average speed andvelocity, and what are the rectangular and polar coordinates of your position?Part BYou have already travelled 30 miles east and then 30 miles north. To get back to the start, you travel 30 miles west in 1 hour and then 30 minutes south in a half hour. What are your average speed and average velocity for the whole rally?

Answer 1

Part A.

30 miles east in half hour next

turn left and drive 30 miles north in 1 hour

The average speed is given by

the total distance divided by the total amount of time.

In this case,

`\begin{gathered} \text{average velocity=}(30+30)/(1.5+1) \\ \text{average velocity=}(60)/(2.5) \\ \text{average velocity=}24\text{ }(miles)/(hour) \end{gathered}`

Now, the position in rectangular coordinates is (30,30)

In polar coordinates, we must find the angle theta and the lenght r.

This can be given as

`\begin{gathered} r=\sqrt[]{x^2+y^2} \\ \end{gathered}`

In our case x=30 and y=30, hence

`\begin{gathered} r=\sqrt[]{30^2+30^2} \\ r=\sqrt[]{2\cdot30^2} \\ r=30\sqrt[]{2} \end{gathered}`

On the other hand, angle theta is given by,

`\theta=tan^(-1)(y)/(x)`

hence,

`\begin{gathered} \theta=tan^(-1)(30)/(30) \\ \theta=tan^(-1)1 \\ hence, \\ \theta=45\text{ degr}ees \end{gathered}`

Therefore, in polar coordinates, point (30,30) is given by

`(r,\theta)=(30\sqrt[]{2},45)`

Part B

Until now, we drove 30 miles east, next 30 miles north. Now, we must drive 30 miles west and 30 miles south.

Hence, we will drive 30+30+30+30=4*30=120 miles. Therefore, the average velocity is

`\begin{gathered} \text{average velocity=}(120)/(1.5+1+1.5+1.5) \\ \text{average velocity=}(120)/(5.5) \\ \text{average velocity=}21.81\text{ }(miles)/(hour) \end{gathered}`

Part A. Velocity.

Velocity is a vector. In this case, we must add to vectors:

vector V1 is given by

`\begin{gathered} v_1=((30)/(1.5),0) \\ \text{which is equal to } \\ v_1=(20,0) \end{gathered}`

Vector V2 is given by

`\begin{gathered} v_2=(0,(30)/(1)) \\ v_2=(0,30) \end{gathered}`

then, resultant vector velocity is

`\begin{gathered} v=v_1+v_2 \\ v=(20,0)+(0,30) \\ v=(20,30) \end{gathered}`