Correction! you should multiply by the mass of the asteroid in part b where it says -sin(theta)*v_impact*r_1. So it should be -sin(theta)*v_impact*r_1*m_asteroid. Sorry if it caused problems
In order to figure out the velocity at impact, you use the energy principle (K + U)i = (K + U)f. Initially, the asteroid's velocity is zero, and it is far enough away for us to consider the gravitational potential energy zero as well. In the final state, the asteroid is nearly touching the...