Seems obvious. If' there's a trick, I didn't get it. My answer, then, is the obvious answer.

Eject sideways so that you don't lose velocity, just change course.

@Sir2u Good job. This way you never negate any component of your velocity vector at any time. Each ejection turns you (roughly over) 45 degrees toward the opposite direction. All momentum is conserved.

@tim wood That doesn't count, since it's only obvious to some people.

Thought of all that, but wouldn't you want to stop once you arrived at base? Absolutely no way to do that by these rules. So you just blast on by at some preposterous speed, no use to anybody except as a battering ram to one alien ship.

Oh wait. Just smack into the alien. Let the springs do their work. Never mind then.

The question only asked how to get to base the quickest. Maybe home base has a circular magnet that slows you down as you go through it. :) Or you can shoot the alien with the fifth projectile at lower energy.

Actually, I just did the math and it looks like by ejecting perpendicularly your final velocity ends up being 4x, compared to if you just cancel out your velocity you end up with a velocity of 6.24x. Even though by ejecting perpendicularly you never negate any of your velocity in any single shot. How about that. :D

Just because aliens have landed doesn't mean it's an "invasion". Endorse multiculturalism (and multispeciesm) and don't spoil your interstellar vacations.

The question only asked how to get to base the quickest. Maybe home base has a circular magnet that slows you down as you go through it. :) Or you can shoot the alien with the fifth projectile at lower energy. — Joseph

Then the answer provided is wrong.
Quickest was is to not do this perpendicular sidetrack. Blow all 5 ejections immediately straight towards home, which gets you there 14x pronto. Stopping once you get there can not be done by the rules, but you seem to have springy bumpers, so crashing is an option.

Hang a Uey.

I think this is what tim wood referred to so I'll leave him as the accepted answer. It's true that ejecting directly opposite from base gets you there quickest, and that's what I calculated, albeit instead comparing with vectors that rotate your motion vector exactly 45 degrees which involves different numbers like square roots of 2.

Sir2u Good job. This way you never negate any component of your velocity vector at any time. — Joseph

As others have already pointed out, the notion that that gets you back faster is a fallacy.

As has been pointed out, just propel yourself directly back toward home.

In general, if you were moving other than directly away from home, you'd want to change your velocity such that the the vector sum of your current velocity and the velocity-change, is toward home.

If you're currently moving away from home, then yes, you should just add propel yourself directly towrard home, in direct opposition to your current motion.

You're drifting through deep space traveling directly away from home base and mission command tells you to return immediately to defend an alien invasion — Joseph

Don't you mean to defend against an alien invasion? Not quite the same thing.

But you're low on fuel and you need to get there as quickly as possible. Your propulsion method is to eject mass in single chunks at high speed.

That sounds like a "mass-driver". It's basically a sort of catapult, maybe electrical or magnetic. They've been proposed for material mined from an asteroid, or a moon. Maybe they'll someday be used for that purpose.

Additionally, mass-drivers have been proposed as a way of changing the velocity of an asteroid on a collision-course with Earth--by launching pieces of the astroid from the asteroid's surface by means of a mass-driver. The obvious advantage is that it uses pieces of the asteroid's rock as reaction-mass.

But of course there could be other ways of using the asteroid's material as reaction-mass, by first pulverizing or vaporizing it, even though that requires additional energy.

Ideally, the mass-driver could just use solar energy, if pieces of the asteroid can feasibly be separated for use with the mass-driver.

But using big chunks of matter as reaction mass--I've never heard that proposed for space-travel. For one thing, big pieces can't be hurled as fast as an ion-stream. Where reaction-mass is scarce, and and is the limiting factor, and energy is plentiful, then the highest exhaust-speed available is desirable.

Ion-exhaust engines have been proposed for space vehicles for some time.

High velocity exhaust requires more energy for a given amount of impulse, but less reaction-mass. Thrust is usually low, because of the higher power-requirement per pound (or Newton, or dyne) of thrust.

Michael Ossipoff