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As the winter passes, the Earth spins as usual around the Sun.On a cold February night just past midnight, a lone astronomerspots an unusual object in the starlit sky. Near the far end of theconstellation of Draco, north of the star HD 91190, there appeareda faint reflective anomaly. After careful observation over the nextfew hours, the astronomer noticed that the object was very close toEarth. Jotting down the coordinates and times, the astronomer cameup with spherical coordinates (astronomers use a different type ofcoordinates, but is ok for this lab):

Feb 28^th -> (x,y,z) = (1.16 x 10⁸ km,6.05 x 10⁸ km , 38.54 x 10⁸ km )

After many days of careful observation, the astronomer foundthat the object was indeed moving! By the middle of May, theastronomer was observing the object at:

May 15^th -> (x,y,z) = (1.05 x 10⁸ km,5.73 x 10⁸ km , 38.28 x 10⁸ km )

Now, with this information, we must decide if this object willcome close enough to Earth that it could collide. There are someother equations that are needed, in particular, the orbit ofEarth:

r = 1.52 x 10⁸ / (1 + 0.0167 * cos(θ)) km

where θ is in degrees, found from Earth's rotation around theSun. Thus, 0⁰ is Dec 21^st, the Winter Solstice, and 180⁰is June 21^st, the Summer Solstice.

Find a set of parametric equations, using the Earth's positionas the origin for each of the object's observations (it will changefor each date).

Given this information, and assume near-linear travel for theunknown object (linear travel does not happen in space, but is okfor this lab)

1. How fast is the object moving?

2. In what direction is the object moving?

3.All of the planets in the Solar System are moving in on anearly flat plane. How long until this object enters into thatplane?

4. Does it seem like this object will hit Earth? Why or whynot?