Your ship is essentially unpowered through te space. An asteroid almost dead ahead has a mass…

Your ship is essentially unpowered through te space. An asteroid
almost dead ahead has a mass about 1000 times the ship’s mass m.
You are moving with speed v0 in +x direction while asteroid is
moving in -x direction at v0/1000. With directional thrusters, you
can make the ship pass close enough to the asteroid so that
asteroid’s gravitational field will deflect the ship’s direction of
motion by some angle θ that depends exactly on how close the ship
passes. ( The figure shows “after” situtation. ) Encounter with
asteroid can be considered “elastic” collision as you are far from
asteroid both initially and finally, so gravitational potential
energy between ship and asteroid is essentially zero both initially
and finally.
a) You want to know whether this encounter might boost the
ship’s speed, helping ship escape. Can it?
b) If so, what is angle θ that yields highest final speed?

FINAL y asteroid ship d