Someone fires a 0.04kg bullet at a block of wood that has a mass of 0.5kg.(the block of wood is sitting on a frictionless surface, so it moves freely when the bullet hits it.) the wood block is initially at rest. the bullet is traveling 300m/s when it hits the wood block and sticks inside it. now the bullet and the wood block move together as one object. how fast are they traveling?

This is an example of a perfectly inelastic collision.
This means your total energy is not conserved before and after the collision, as some it is lost in deforming the material and heat. 
The only conserved quantity is momentum (which is always conserved).

We'll compare the momentum before and after the collision.
Recall momentum is defined as 
Also recall momentum is a vector, as it depends on velocity, not speed.

Before the collision, the piece of wood is at rest, so the initial momentum is equal to -only- the momentum of the bullet.

In which  is the total momentum before collision, is the mass of the bullet, and is the initial velocity of the bullet (300 m/s).

After the collision, we have the momentum of the bullet and wood moving together with an unknown velocity -both wood and bullet have the same velocity, so we can factor out the summation of their masses-.

In which is the total momentum after collision, is the velocity we're looking for, and is the mass of the wood.

Now we apply conservation of momentum:



Now we can easily solve for :