Two objects, \(m\) and \(M\), with initial velocities \(v_0\) and \(U_0\) respectively, undergo a one-dimensional elastic collision. Show that their relative speed is unchanged during the collision. (The relative velocity between \(m\) and \(M\) is defined as \(v-U_{-}\))

The concept of conservation of momentum is at the very heart of analyzing elastic collisions in physics. Momentum, the product of an object's mass and its velocity, is a vector quantity, meaning it has both magnitude and direction. According to the law of conservation of momentum, the total momentum of a closed system remains constant if no external forces are acting on it.

In the context of a one-dimensional elastic collision, such as the interaction between two objects denoted by masses m and M with initial velocities v_0 and U_0 respectively, the conservation of momentum can be mathematically expressed as:
\[m v_0 + M U_0 = m v_f + M U_f\]
This equation tells us that the combined momentum of both objects before the collision is equal to their combined momentum after the collision. In the steps provided by the textbook solution, this principle allows us to link together the initial and final velocities of the two objects involved in the collision.

Understanding conservation of momentum is critical as it provides a framework to predict the outcome of collisions without knowing the intricate details of the interaction forces between the objects.

Another pivotal concept in elastic collision analysis is the conservation of kinetic energy. Kinetic energy is the energy an object possesses due to its motion and, like momentum, is conserved in the absence of external forces in elastic collisions.

The formula for kinetic energy is:
\[\frac{1}{2} m v^2\]
where m represents the mass and v the velocity of an object. For a collision between two objects, the conservation of kinetic energy can be written as:
\[\frac{1}{2} m v_0^2 + \frac{1}{2} M U_0^2 = \frac{1}{2} m v_f^2 + \frac{1}{2} M U_f^2\]
This ensures that the total kinetic energy before the collision equals the total kinetic energy after the collision. The solution steps leverage this conservation law to establish a relationship between the squares of the initial and final velocities. Importantly, this conservation only holds true for elastic collisions—in inelastic collisions, some kinetic energy is converted to other forms of energy, such as heat or sound.

The concept of relative velocity is especially important when studying collisions. Relative velocity represents the velocity of one object as observed from another object. In the case of our one-dimensional collision problem, the relative velocity between two objects m and M is given by:
\[v - U\]
where v and U are the velocities of m and M respectively. For elastic collisions, one intriguing result is that the relative speed—meaning the magnitude of the relative velocity—before and after the collision remains unchanged.

The textbook's step-by-step solution demonstrates this by showing that when you calculate the difference of the final velocities v_f and U_f and compare it to the initial state v_0 and U_0, they equate, thereby proving that:
\[v_f - U_f = v_0 - U_0\]
This result has far-reaching implications in physics, indicating that the approach and separation speeds of two objects in an elastic collision are consistent, which can be used to predict future positions and velocities post-collision. This understanding of relative velocity is not only crucial for problem-solving in classical mechanics but also in various practical applications, such as in the analysis of vehicle crashes and in astrodynamics.