NMS 1.3- A kid is on a carrousel, describing a circumference of (2m radius), completing a lap every 8 seconds. You can see the trajectory in the drwaing seen from above. Considering the kid only stays for one lap:
a - adopt a coordinate system centered in the reference point O. Draw and determine the position vectors of the kid in points: A, B, C, D and E. b- Determine and portrat the displacement vectors between: A and E; A and D; A and C; A and B. c- Determine and portray (in other scheme) the average speed vectors between:  A and D; A and C; A and B. d- (Vectorial kinematics) e- Find the displacement vector and the average speed between: points A and D, A and C, in this new system, and compare the results with the ones from: b- and c- . f-Represent the instant speed vectors in points A, B and C. Do they depend on the reference system we take? Wich is their intensity?

Long as poor man´s hope... but we have no choice. Let´s begin.

 î and ĵ are vectors of modulus 1, also called "Versors", their objetive is to join the vectorial characteristics of a number and tell where it is pointing to. Usually it is portrayed with a little hat or a little comma above ( replacing the typical i and j ). Versor î has the same direcction and sense in the x axis, as the ĵ versor in the y axis.

b - The displacement vector, Δr_AE, for example, is the vectorial subtraction between the rear position vector and it´s previous one, r_E — r_A; To find it, you must subtract component by component. It´s easy, just look.

Δr_AE =  r_E — r_A

Δr_AE = (— 2 m î + 0 m ĵ ) — (0 m î — 2 m ĵ ) = — 2 m î + 2 m ĵ

I just do the same with the other ones.

Δr_AD = 0 m î + 4 m ĵ

Δr_AC = 2 m î + 2 m ĵ

Δr_AB = 1,41 m î + 0,59 m ĵ

I portrayed it in green, just like the others. I will portray them twice. The one in the left is the geometric operation: to subtract two vectors, you just have to join them together by their ends. The subtract vector always have it´s origin in the first vector and ends on the second one.

In the one at the right, I portrayed them just how I got them by doing it analitically, then, they appear centered in the origin of the reference system. Just look at it, there you have them.

c -The average speed is:

v_m = Δr / Δt

To analytically obtain the average speed vector (vm) you divide each component of the Δr vector by the time interval, wich is different for each position, but it´s easy to calculate, because if it taked 8s to finish an entire lap, it takes 4s to finish half a lap, and so on.

e -If you do the math, you will notice that the results (and portrayals) are the same we had before.

f - The instant speed I obtain it dividying the displacement (measured over the trajectory) with it´s corresponding time interval. For example: to finish one lap, that is, 12.56 m (3,14 times the diameter), it takes 8 s. Then, the instant speed is: v = 1.57 m/s. I made two drawings: one in sity, that is, the carriage; The other one, in a speed chart.

Once more, you can nottice that the scales I use for the lenghts have nothing to do with the scales I use in the speed charts, and that´s the way to be.

CHALLENGE: obtain all the vector´s modulus in this problem. Compare them between each other, and find the angles they form with the x axis. And then, yes, go have some vacations to the Canary islands.