Hello! This is my first achievement guide so go easy on me. Feedback appreciated!
I actually found this achievement fairly straightforward so I wanted to share my method here.
First, boost most of the way to the mothership using RT. The chasing enemy ships shouldn't do you any damage if you're moving at top speed. When you reach the target you'll see two golf balls on the top of the mothership. As you come within shooting range, get your crosshair on the closer golf ball, and hit LB, LT and A, holding all three down. This will make you brake, roll and shoot all at the same time. By slowing down, you should be able to do enough damage to destroy the first golf ball in one pass. You'll also take minimal damage from the turrets and chasing ships.
As soon as you see the first golf ball explode, release everything and boost away to the other side of the ship using RT. Again, the chasing enemy ships and mothership turrets shouldn't do you much damage, especially if you change direction from time to time. Once you've put some decent distance between you and the mothership, loop back around and target the second golf ball using the same method. I like to use LT and LB together to get a tight turn.
Once the second golf ball is destroyed, boost away again but this time to the underside of the mothership. There is a larger third golf ball under there. Again, put some decent distance between you and the mothership before looping back around and targetting the golf ball whilst braking and rolling. You should also be able to destroy this golf ball in one pass.
If you're unable to destroy any of the golf balls on your first pass, simply boost away and loop back. Target the golf ball again and use the LB, LT & A combination another time.
Once the three golf balls are destroyed, boost back up to the top side of the mothership and charge straight at the tower between the two destroyed golf balls. Approach from the front side of the mothership and that should do the trick.
Many thanks to the original poster whose method I initially used. Hopefully the video makes everything clear!