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 circe orbit at once equal to $2\pi \approx 6.28$ days

b) Velocity (in AU per day):

$$v_x(t) = \frac{dx}{dt} = 5 \cos(t) \, \quad v_y(t) = \frac{dy}{dt} = 2 \cos(2t)$$

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

c) At time $t= \pi/2, \, x = 5,\, y= 0, \, z=0$

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

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