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 = 5 sin(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
?
Expert's answer

Answer 1

a) The highest period is

$$T=2\pi \approx 6.28\ days$$

 b)

$$\vec{v}=\left(\frac{dx}{dt},\frac{dy}{dt},\frac{dz}{dt}\right)$$

$$\vec{v}=\left(5\cos{t},2\cos{2t},0\right)$$

The magnitude is

$$v=\sqrt{(5\cos{t})^2+(2\cos{2t})^2}$$

c)

$$\vec{r}=\left(5,0,0\right)\ AU$$

The distance from the first star is

$$x-x_0=5-4=1\ AU$$

The distance from the second star is

$$x-x_1=5-(-4)=9\ AU$$