In a frame at rest with respect to the billiard table, a billiard ball of mass \(m\) moving with speed \(v\) strikes another billiard ball of mass \(m\) at rest. The first ball comes to rest after the collision while the second ball takes off with speed This shows that momentum is conserved in this frame. (a) Now, describe the same collision from the perspective of a frame that is moving with speed \(v\) in the direction of the motion of the first ball. (b) Is the momentum conserved in this frame?

The momentum conservation principle is a fundamental law in physics that states that the total momentum of a closed system remains constant if no external forces are acting on it. Momentum, the product of an object's mass and velocity, is a vector quantity, meaning it has both magnitude and direction.

Let’s consider our billiard ball example. Initially, one ball moving with momentum of magnitude mv and the other at rest (momentum is zero), combine to a system momentum of mv. After the collision, the system's momentum still sums to mv, with the previously stationary ball now moving, demonstrating that momentum was indeed conserved. This is an example of an elastic collision, where kinetic energy is also conserved.

In any inertial frame of reference—meaning one moving at a constant velocity without acceleration—the physics is the same. Momentum conservation applies no matter how the observer is moving relative to the system, as long as external forces are not involved. This concept also helps students understand that while individual velocities may change as per the frame of reference, the overall physics of the system does not.

An inertial frame of reference is a perspective from which an observer is either at rest or moving with a constant velocity. This means no net external force is causing the observer to accelerate. In these frames, Newton's first law of motion, which states that an object will remain at rest or in uniform motion unless acted upon by an external force, holds true.

In our exercise, both the table’s frame and the moving frame are inertial – one is stationary and the other moves with uniform speed. Physics laws, specifically momentum conservation, apply identically in both. Confusion often arises when the velocities of objects appear different due to the relative motion of the frame of reference. However, it's important to emphasize to students that this does not affect the validity of the laws of physics within that frame.

An elastic collision is one in which both momentum and kinetic energy are conserved. In the case of our two billiard balls, both share the same mass, and the collision results in an exchange of velocities without any loss of kinetic energy to sound, heat, or deformation. This is an ideal scenario often used in physics problems to simplify the analysis.

After the collision, the initially moving ball comes to rest, meaning all its kinetic energy has been transferred to the second ball, which now moves with the same speed as the first one did before the collision. When considering elastic collisions, it’s essential for students to check both momentum and kinetic energy before and after the event to ensure that the collision fits the definition of being 'elastic'.

Relative velocity is the measure of the velocity of one object as observed from another object or frame of reference. The concept is crucial when analyzing any system involving motion because it helps us to understand the motion of objects relative to a particular observer.

Returning to our exercise, from the billiard table's frame, the second ball's velocity is v after the collision. However, from the frame that is moving at v, the second ball's velocity is 0 post-collision - it's at rest relative to this moving frame of reference. This difference is because we are observing the motion from different perspectives, yet despite these differing observed velocities, the underlying physical law (momentum conservation) remains steadfast.