Question

You are the captain of a spaceship that is circling through a binary star system. Due to the gravi-

tational 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

?​

Answer 1

Given :

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:

T=2π= 2x3.14=6.28days

b) Find an equation that calculates the velocity v(t) of your spaceship at a given time t

→

v= (dx/dt, dy/dt , dz/dt)

=(5 cost  ,2 cos2t, 0)

The magnitude =√((5cost)² +(2cos2t)²+0²)

c)The two stars are positioned at the points (4, 0, 0) and (−4, 0, 0)

x=5 , y=0 z=0

→

r=(5,0,0) AU

The distance from the first star is

x-x0=5-4=1 AU

The distance from the second star is

x-x1=5-(-4)=9AU