A skier is moving straight on a slope with an inclination such that the

skier accelerates with constant acceleration $$a of magnitude $0\mathrm{.}5$ m$/$s$2.$

i$)$ Draw a $($smart$)$ Cartesian coordinate system that you will use to study the motion of the skier.

ii$)$ In the chosen Cartesian coordinate system, sketch the skier$$s position vectors, $$r$1$ and $$r$2,$ when the skier has covered the distances k$1$ and k$2.$ Draw also the displacement vector corresponding to these distances. In addition, sketch the velocity and acceleration vectors, $$v$1,$v$2,$a$1,$a$2$ at the time the skier has covered the distances d$1$ and d$2.$ Comment whether each of these vectors is parallel$/$orthogonal$/$neither$-$parallel$-$nor$-$orthogonal to the trajectory.

iii$)$ Determine by what amount the quantity $|$v$1|2-|$v$2|2$ increases when: the acceleration is increased by a factor $2,5,10,20,$ and the distance difference s$1-$s$2$ is unchanged, e$.$g$.$ s$1-$s$2=200$ m; the distance difference s$1-$ s$2$ is increased by a factor $0\mathrm{.}5,0\mathrm{.}2,0\mathrm{.}1,0\mathrm{.}05,$ and the acceleration is unchanged. Can you state with respect to which quantities in your expression, $|$v$1|2-|$v$2|2$ is directly or inversely proportional? And if you increase $|$a$|$ by a factor N$,$ how much do you have to increase $|$a$|$ to keep $|$v$1|2-|$v$2|2$ constant?

iv$)$ Probably you have already done the following for iii, but if not, determine the quantity $|$v$1|2-|$v$2|2$ in terms of a generic acceleration $$a and a generic distance difference s$1$ and s$2.($Hint: with our tricks on derivatives, the solution is much faster. Try to re$-$express the acceleration d$$v$/$dt in terms of ds$/$dt$.)$

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v$)$ Write the units of every quantity appearing in the equality you found in iii. Moreover check whether the units on the left hand side and right hand side of that equality are the same.

vi$)$ Evaluate the components of $$v$2($in the coordinate system of point i$)$ and the speed at the distance s$2$ if you know that $|$v$1|=10$km$/$h$,$ s$1=100$ m and s$2=0\mathrm{.}5$ km$.$

vii$)$ With the data provided in vi$,$ determine the distance s$3$ at which the speed of the skier is $100$ km$/$h$.$

viii$)$ With the data provided in vi but s$2$ still unfixed, sketch a plot showing how $|$v$2|2$ depends on s$2.$ In this plot, what is the value of the slope at a generic s$2$