A simple mechanics problem The problem says:

A bullet hits a block kept at rest on a smooth horizontal surface and gets embedded into it. Which of the following does not change?
A. Linear momentum of the block
B. Kinetic energy of the block
C. Gravitational potential energy of the block
D. Temperature of the block.

My teacher said that the answer would be C. If we conserve linear momentum the linear momentum of the block would change, and so would the kinetic energy. A part of this would be spent as heat would change the temperature of the block. So C would be the answer.
But how can we conserve linear momentum here? There is a normal contact force acting on the block and thus, there's an external force on the system!
 A: 
There is a normal contact force acting on the block and thus, there's an external force on the system!

There is no external force on the system rather there is an external force on block land bullet.  Hence momentum of block and bullet is not conserved but momentum of system is conserved.
A: There are two vertical forces acting on the block: The normal force and the weight of the block. The vertical net force is zero so there is no change in the vertical component of the momentum of the block. On the other hand, the bullet exerts a horizontal force on the block (and so does the block on the bullet). So the horizontal component of the block's momentum changes from zero to some non zero value. The fact that the momentum changes means that the velocity changes and so does the kinetic energy.
The horizontal momentum of the system block and bullet may be conserved but this is not an important issue for this problem. The question is about the block.
A: The block is on a horizontal surface. The normal force acts in a vertical direction. Since you are considering motion in the horizontal direction, there is no net force.
A: Always remember,
When things stick to each other, energy is not conserved but the momentum is conserved.
Your system is made of bullet and the block. The bullet puts some force on the block. So, no external force is added to the bullet and the block which together are your system. The force between the bullet and the block is not an external force.
Here is an example in which linear moment is not conserved. Pendulum. When you keep the pendulum horizontally and then let it go and it will start to swing because of the gravity. Here gravity is an external force on the pendulum system and the momentum is not conserved.
A: Momentum is conserved if there is no NET external force acting on the system. The normal force on the block is balanced out by the force of gravity on the block.
You may say, "what about the force of gravity acting on the bullet -- does that cause an external force?" It does, however, if you consider the speed of the bullet right before the collision which takes up only a small amount of time, the work done by gravity on the bullet will be so negligible that total momentum can be considered constant.
Also, the normal force and force of gravity act only in the up/down $y$ direction. If you consider only the $x$ momentum, then it must be conserved since no force acts in the $\pm x$ direction.
A: 
There is a normal contact force acting on the block and thus, there's an external force on the system!

This normal force is balanced by the weight of the object. Until the block leaves the surface, the net vertical force is zero. On the contrary, you applying the conservation of momentum along the horizontal direction, therefore vertical forces does not matter.
And the important point is that your question is asking about the block only, not the whole system. As a system, definitely the momentum is conserved. Though you can see block is moving after the collision which was at rest before. So it has gained some velocity. Doesn't that imply the block has increased its momentum?
