What happens if you slow down a moon? Let's say you'd put a whooping big rocket engine on the surface of the moon, and slow down its orbital velocity.
Not to a complete stop, mind you. Just a bit slower than it was going before (a couple of m/s?).
Intuitively (i.e. without doing the maths, or even knowing the maths involved), I'd say it would start spiralling towards earth, eventually crashing into it.
I was wondering about this due to this YT video about Kerbal Space Program that talks about how much it would take to stop a moon. I figured that you don't need that much delta-V, that a much smaller amount of slow-down would already (eventually) de-orbit a moon already.
Is that correct?
 A: To shine a light on Emilio's comment and develop Fabrice's answer, what will happen is the following. Given that the change in velocity can be considered instantaneous relatively to the period, the moon will find itself in a new orbit where its position (just after the slow down) is the apoapsis (the furthest point from the focus where lies the Earth) of a new elliptical orbit. As you guessed the moon will then gain speed as it comes closer to the Earth (but not spiraling down) in this new elliptical orbit until it reaches its maximal velocity at the periapsis of the orbit, the greater the slow down of the moon, the smaller the periapsis. Now if this slow down is too big the moon can effectively crash into the earth.
See Hohmann transfer for more detail :https://en.wikipedia.org/wiki/Hohmann_transfer_orbit
A: In fact, our moon is slowing down very gradually due to tidal effects and (counter-intuitively) that has caused the moon to move very much farther away from the earth since their creation.  According to Kepler's laws of orbital motion a body in orbit sweeps out the same area in the same amount of time.  As the motion slows down the body moves farther away to provide the same swept area per unit of time. If you want to see a good discussion of this view the first and second of Richard Feynman's Messenger Lectures on YouTube.
Tides slow down orbiting bodies by converting the energy of motion to heat due to moving the masses of oceans AND continents.  Friction and compression/release of molecular bonds inefficiencies converts the water and land motions to heat energy.
Adding more:  Emilio is right.  What is occurring is that the Earth's angular momentum of rotation (day and night) is being transferred to the moon's orbital speed.  This occur's because the oceans bulge out and then are slowed by friction and blocked by the continents.  The friction and blockage keep the ocean bulges from synchronizing with the moon's orbit.  The resulting off-set bulges exert a gravitational torque on moon, transferring energy.  The moon speeds up and moves further away and the earth's rotation slows down.  
A: It would not be spiraling, it would be on a different elliptical orbit, probably less circular. All 2-(solid)-body gravity equilibriums are ellipses. Ellipse is just the constantly renewed balance between free fall and inertia of motion in the current direction. Spiraling requires continuously losing energy (i.e. being sucked by something, in your case).
