Momentum — common mistakes

Always draw clear 'before' and 'after' diagrams. Define a positive direction and stick to it. When applying Newton's Law of Restitution, remember it relates the relative speeds. A common mistake is to forget the coefficient $e$ or to mix up the velocities. Writing 'CoM' and 'NLR' next to your equations is good practice to show the examiner your method.

Momentum ($p=mv$) is a vector quantity that is always conserved in a closed system (no external forces). Kinetic energy ($KE = \frac{1}{2}mv^2$) is a scalar quantity that is only conserved in perfectly elastic collisions ($e=1$). In most collisions, some kinetic energy is lost to heat, sound, or deformation.

Yes. Because momentum is a vector, its sign depends on the direction of motion relative to a chosen positive axis. A negative momentum simply means the object is moving in the opposite direction to the one you've defined as positive.

In a collision between two particles, there are typically two unknowns (the two final velocities). To solve for two unknowns, you need two independent equations. The Conservation of Momentum provides the first equation, based on Newton's laws. The Law of Restitution provides the second, which accounts for the energy loss or 'bounciness' of the specific materials involved in the collision.