$4$ A moon orbits a planet, that orbits a star, that orbits a black hole $[20\%]$

The black hole is at the origin of coordinates. At $l=0$ the star is at the point $(1,0),$ the planet at the point $),$ and the moon at the point $(1\mathrm{.}15,0).$ The period of the star is $T=2$; this means that the star completes one orbit after one pertiod, i$.$e$.,$ at $t=2=T.$ The planet completes an orbit around the star when $t=2\frac{}{10},$ and the moon completes an orbit around the planet at $l=2\frac{}{20}.$

$($by hand$)$ Build parametric curves that describe the motion of the star, planet and moon. Use a computer or calculator to graph the parametric curves from $t=0$ to $L=2.$

Compute and compare the arelengths of the three parametric curves from $t=0$ to $=2.$ The arclength formula is

$L={\displaystyle \underset{t_0}{\overset{t_4}{}}}\sqrt[\phantom{2}]{x^{\text{'}}(t{)}^2+y^{\text{'}}({)}^2}dt$

$($by hand$)$ How would you calculate the average spocds of star, planet and moon? Write down a mathematical expression for the average speod and justify it$.$ Evaluate your formula for star, planet and moon and rank them? Who is the fastest? Explain.

Figure $1$: Snippet of the trajoctory of the star$-$planel$-$moon system