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The problem is as follows:

In the figure from below it is shown an observer who has put himself at the center of the coordinate system. He sees an object moving in a circular trajectory. If the average speed between $A$ and $B$ is $\left ( -2\hat{i}+\hat{j} \right )\frac{m}{s}$ and its position on point $A$ is $5\hat{i}\,m$. Find on $\frac{rad}{s}$ the average angular velocity between $A$ and $B$ if the time the object takes to get from $A$ to $B$ is $4\,s$.

Sketch of the problem

The alternatives on my book:

$\begin{array}{ll} 1.&0.35\hat{k}\\ 2.&0.5\hat{k}\\ 3.&0.55\hat{k}\\ 4.&0.6\hat{k}\\ 5.&0.75\hat{k}\\ \end{array}$

For this particular problem I'm stuck at how to use the information given the average velocity and the position. However I recall that when the word average is mentioned it mean this formula?

$\overline{v}=\frac{\vec{r}}{\Delta t}$

But other than that I'm not sure if it applies in this situation. Can somebody offer some help with this question?.

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After 4s the object has moved $-8i+4j$ to $(-3,4)$ and so it has travelled through an angle of $\pi - tan^{-1}(\frac{4}{3})$ in 4s. This is $0.55 s^{-1}$ and so the answer is (3.).

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  • $\begingroup$ I am stuck at understanding your solution. It looks a bit rushed. Can you please explain the steps in the middle so I can understand what you did and why you did it?. I'm assuming that you used the given time to obtain the second position and from then you used the given coordinates to get the angle the next part was to find the traveled angle and with time you found the speed which was what was being asked right? But why on earth did you put $s^{-1}$?. Wouldn't it be $\frac{rad}{s}$?. Or should I understand that by definition rad is dimentionless?. $\endgroup$ – Chris Steinbeck Bell Oct 27 '19 at 21:02
  • $\begingroup$ where did you obtained the $\left(-3,4\right)$? I'm stuck at this part?. As it stands now it looks as if a magician took a rabit from a hat. Could it be that is from the subtraction of the second coordinate and the first one?. Or in this case we are referring to the resultant vector which would be the sum of the given vectors?. Am I right with this?. Although I think that the displacement is the subtraction mind giving me a hand with this definition? $\endgroup$ – Chris Steinbeck Bell Oct 27 '19 at 21:09
  • $\begingroup$ It starts at (5,0) and moves 4x(-2,1) to (-3,4). $\endgroup$ – S. Dolan Oct 27 '19 at 21:18
  • $\begingroup$ Note that (-2,1) is the average velocity and therefore you only have to multiply by the time to find the displacement. Also, yes to your query about units - the SI units are $s^-1$ but you ca use rads per sec if you wish. $\endgroup$ – S. Dolan Oct 27 '19 at 21:22
  • $\begingroup$ I believe your answer could be benefited by adding the fact that the position is not directly given, but instead the average velocity. And we can obtain the displacement between the two by multiplying the time to the speed. Since one position is given we can also find other position by that subtraction. From that we can obtain the angle and with the radius, the length and the time we can obtain the speed. Am I right with what I described follows your logic?. $\endgroup$ – Chris Steinbeck Bell Nov 1 '19 at 23:25

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