More physics, yawn...
Sliding downhill converts potential energy (the energy the car has because it's higher at the start of the slide than what it is at the end- its "position") into kinetic energy (i.e. its motion- it's going faster at the bottom of the hill than at the top). Like a roller coaster, you've traded the energy of position (higher to lower, so more potential energy at the top of the hill than at the bottom) for the energy of motion (going faster at the bottom).
In physics, it's momentum (mass times velocity) that's conserved, unless an external force is applied. You can change momentum only by changing how much mass an object has, or how fast it's going.
By the way, that's how recoil works: Just as the bullet exits the muzzle the bullet and the gun have the same amount of momentum, but in opposite directions. The bullet has small mass and so is going much faster, the gun is way heavier and so goes much slower, in the opposite direction. Momentum is conserved- the sum of it all before the shot is fired is zero (no velocities) and so must add up to zero afterward. The MV for the bullet and MV for the gun are equal in magnitude but in opposite directions, so they add up to zero because velocity is plus in one direction and negative in the opposite direction- it's a "vector".
When a bullet strikes something its momentum (or the momentum of all of its pieces if it fragments) is reduced by an external force, which is applied to it by the target. it's in providing this force that a target is damaged. The change in momentum an object undergoes is calculated by multiplying the force on it by the length of time that force is present. This is called "impulse" in physics. If the target is soft or fragile, providing this force will damage it; if the target is hard it may not- see the gif mello2u posted, where a huge force is required because it's applied only for as long as the bullet is pushing on the wall, a very short time. This is why air bags help reduce impact injuries- they spread out the time the force is applied, which reduces the force, hopefully to a level below that which would produce serious injury.
Energy is the more common term when the effects of bullets are measured. In physics, energy can be changed by "work", which has a very narrow meaning in physics. The amount of kinetic energy an object has is equal to 1/2 its mass times the square of its velocity. A bullet has high kinetic energy (lots of velocity); reducing that energy is done in the target by applying a force in a direction opposite that of the velocity, for some measured distance. The product of force times the distance through which it acts (how "work" is calculated in physics) has to equal the kinetic energy if the bullet is to stop. That's why hollow points and soft nosed bullets are used- they spread out so that a larger force is required which keeps the distance short, and inside the target so all of the bullet's energy is used up in the target (its end velocity is zero) which maximizes the damage it does. A bullet that doesn't expand doesn't meet a lot of force inside the target because it's still streamlined, so the distance to stop it is longer, often far enough that the bullet passes all the way through the target. This means the target didn't get all the bullet's energy because the bullet still has some velocity left after exiting.
It's in providing this force that a target is damaged- sending a high speed projectile at something transfers energy to it which it must deal with, and that can be very destructive.
The recoil energy of the gun is much less than the muzzle energy of the bullet because energy depends on the SQUARE of velocity and there's a huge difference between the velocity of the bullet and the gun after the shot. In absorbing the recoil, both the force (how hard it hits you) and the distance it moves (where's the muzzle pointing after it's all over) determine the recoil experience. If the gun is light (a handgun) then its recoil velocity will be higher than the same cartridge in a rifle.
