Answer to Question #89388 in Astronomy | Astrophysics for Peace Adejumobi

Question #89388
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
?
1
Expert's answer
2019-05-15T10:11:30-0400

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"



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