In a totally inelastic collision between two equal masses, with one initially at rest, show that half the initial kinetic energy is lost.

In physics, particularly in the study of mechanics, the principle of conservation of momentum is a fundamental concept that states that the total momentum of a closed system remains constant provided no external forces act upon it. Momentum, which is given by the product of an object's mass and its velocity, is a vector quantity, meaning it has both magnitude and direction.

To understand the conservation of momentum in the context of collisions, imagine two pool balls colliding on a table. The sum of their momenta before the collision must equal the sum of their momenta after the collision if no external forces interfere. In the case of inelastic collisions, where objects stick together and move as one after the impact, the combined mass of the resulting object moves with a velocity that ensures the total momentum is preserved. Mathematically, if two objects with masses and velocities represented by \(m_1, v_1\) and \(m_2, v_2\) collide inelastically, the conservation of momentum can be represented as \(m_1 v_1 + m_2 v_2 = (m_1+m_2) V\), where \(V\) is the common velocity after the collision.

During an inelastic collision, the principle of conservation of kinetic energy does not hold true, unlike the conservation of momentum. Kinetic energy is the energy an object possesses due to its motion, and it can be calculated using the formula \(KE = \frac{1}{2} m v^2\), where \(m\) is the mass of the object and \(v\) its velocity.

When two objects collide inelastically, they lose some of their kinetic energy—the energy is transformed into other forms, such as heat or sound. This loss of kinetic energy often results in a decrease in the relative speed of the objects post-collision. The problem we're discussing illustrates a significant aspect of inelastic collisions: when a moving object collides with another object of equal mass at rest, up to half of the initial kinetic energy can be lost, depending on the specific conditions of the collision.

Analyzing inelastic collisions requires considering both momentum and kinetic energy. In our example, a stationary object with mass \(m_2\) is struck by a moving object with mass \(m_1\) and velocity \(v\). After the collision, the objects stick together, resulting in a perfectly inelastic collision.

To analyze the outcome, we first apply the conservation of momentum to find the combined velocity \(V\) of the new, single object comprised of both original masses. The energy analysis then follows; we compare the initial and final kinetic energies to find out how much energy was lost due to the collision. Despite the momentum being conserved, there's a clear distinction in the energy before and after the collision—which, as in the given problem, concludes with a reduction of 50% in kinetic energy, demonstrating a core difference between elastic and inelastic collisions.

Understanding the interaction between momentum and energy in collisions is integral to grasp the nuances in different types of collision scenarios. Momentum, always conserved in collisions, dictates the motion of objects post-impact. Kinetic energy, however, may not remain constant if the collision is inelastic.

In an elastic collision, both momentum and kinetic energy are conserved. The objects collide and rebound without a permanent shape change or any energy transformed into other forms. On the other hand, in inelastic collisions, only the momentum is conserved, and kinetic energy is not. A portion of kinetic energy is typically converted to other forms of energy, resulting in the system having less kinetic energy post-collision. The numerical analysis in our exercise illustrates these principles with an example of a perfectly inelastic collision where kinetic energy is halved, manifesting the energy conservation disparity between elastic and inelastic collisions.