Why are we applying conservation of momentum in the following question? A bullet of mass 0.01kg is fired horizontally into a 4kg wooden block at rest on a horizontal surface. The coefficient of friction between the block and the surface is 0.25. The bullet remains embedded in the block and the combination moves 20m before coming to rest. With what speed did the bullet strike the block?
To get the answer, we are supposed to apply the conservation of momentum.
Why? Isn't the frictional force acting as an external force in the given situation?
 A: As mentioned in the comment section, there are two processes in action here.

*

*Bullet hitting the block

*Motion after the impact

And the most important point is that friction plays its role only after the motion starts since it is its job() to oppose relative motion between surfaces in contact. And in your question, the relative motion starts only after the impact.
During the first instant of the impact, the block was considered to be at rest.
So there is no friction in action in the first part and hence you can blindly apply conservation of linear momentum for the impact.
As mentioned in the comment below by Luke Pritchett, it is important to note that it is true only if we assume the momentum transfer process is completed instantly after the collision.
Hope it helps .
A: The force of friction is an external force on the block.  During the short time of the impulsive impact $\Delta t$ the change in momentum due to the force of friction is $F_{fric} \enspace \Delta t = m_{total} \enspace \Delta v$ where $m_{total}$ is the mass of the block and the imbedded bullet, and $\Delta v$ is the change in velocity of the block\bullet due to the force of friction $F_{fric}$.  Since $\Delta t$ is very short, $\Delta v$ is negligible. After the impact, the force of friction will then retard the motion of the block and bullet.
If it assumed the block/bullet is a rigid body, the force of friction reduces the kinetic energy of the block/bullet with no "heating" effects, since for a rigid body there can be no dissipation of energy within a rigid body.  In reality the block/bullet is not perfectly rigid and "heating" occurs.
A: When you lose energy (because of friction) you cannot use conservation of energy, but you can still use conservation of momentum. Common examples are when the two bodies stick to each other (like the one you mentioned)
You can't use conservation of momentum WHEN THERE IS AN EXTERNAL FORCE acting on the system. Friction is a five between the object and is not external so it does not mess up the conservation of momentum.
An example of this type of motion is pendulum.
