Question

You are the captain of a spaceship that is circling through a binary star system. Due to the gravitational forces and the rocket engines, the orbit of your spaceship looks like that:

The position of your spaceship (in AU) at the time t (in days) is given by:
x= 5sin(t) y= sin(2t) z= 0
a) How long does it take your spaceship to circle the orbit once?
b) Find an equation that calculates the velocity v(t) of your spaceship at a given time t.
c) The two stars are positioned at the points (4,0,0) and (-4,0,0): what is the distance of your spaceship to the stars at the time t = π/2 ?

Answer 1

a)

Period of time to circle orbit at once equal to

$$2π≈6.282\pi \approx 6.282π≈6.28 days$$

b)

$$v(t)= \sqrt{ v_x(t) ^2+ v_y(t) ^2} (1)$$

where

$$v_x(t)= \frac {dx} {dt}=5\cos(t)$$

$$v_y(t)= \frac {dy} {dt}=2\cos(2t)$$

We got:

$$v(t)= \sqrt{25 \cos(t) ^2+ 4\cos^2(2t)}$$

c)

x=5, y=0, z=0

distance to the star at the point (4,0,0) is equal to 1 a.u., distance to the star at the point (-4,0,0) is equal to 9 a.u.