solve the following problems $1)$ Two meteoroids are heading for earth. Their speeds as they cross the moon's orbit are $2\mathrm{.}0$ km$/$s$.$ The second misses the earth by $4000$ km$.$ What is its speed at its closest point? $2)$ A new planet is discovered orbiting the star Vega in a circular orbit. The planet takes $55$ earth years to complete one orbit around the star. Vega's mass is $2\mathrm{.}1$ times the sun's mass. What is the radius of the planet's orbit? Give your answer as a multiple of the radius of the earth's orbit. $3)$ The two stars in a binary star system have masses $1\mathrm{.}810^30$ kg and $6\mathrm{.}310^30$ kg$.$ They are separated by $2\mathrm{.}510^12$ m$.$ A$)$ What is the system's rotation period, in years? B$)$ What is the speed of the lightest star? C$)$ What is the speed of the heaviest star? $4)$ In $2000,$ NASA placed a satellite in orbit around an asteroid. Consider a spherical asteroid with a mass of $1\mathrm{.}5010^16$ kg and a radius of $8\mathrm{.}20$ km$.$ A$)$ What is the speed of a satellite orbiting $4\mathrm{.}70$ km above the surface? B$)$ What is the escape speed from the asteroid? $5)$ Consider a neutron star with a mass equal to the sun, a radius of $19$ km$,$ and a rotation period of $1\mathrm{.}0$ s$.$ A$)$ What is g at the surface of this neutron star? B$)$ A stationary $1\mathrm{.}0$ kg mass has a weight on earth of $9\mathrm{.}8$ N$.$ What would be its weight on the star? C$)$ How many revolutions per minute are made by a satellite orbiting $1\mathrm{.}0$ km above the surface? D$)$ What is the radius of a geosynchronous orbit about the neutron star? $6)$ Two Jupiter$-$size planets are released from rest $6\mathrm{.}010^10$m apart. What are their speeds as they crash together?