In an American football game, a 90.0-kg fullback running...

In an American football game, a 90.0-kg fullback running east with a speed of $5.00 \mathrm{~m} / \mathrm{s}$ is tackled by a $95.0-\mathrm{kg}$ opponent running north with a speed of $3.00 \mathrm{~m} / \mathrm{s}$. (a) Explain why the successful tackle constitutes a perfectly inelastic collision. (b) Calculate the velocity of the players immediately after the tackle. (c) Determine the mechanical energy that disappears as a result of the collision. Account for the missing energy.

Key Concepts

Perfectly Inelastic Collision

In a perfectly inelastic collision, the colliding objects stick together and move as one mass after the collision. This type of collision is characterized by the conservation of momentum while kinetic energy is not conserved. The kinetic energy that appears to be lost is converted into other forms of energy, such as heat and sound, as well as energy used in deformation of the objects. This concept is particularly relevant in collision problems where the objects merge post-impact.

Conservation of Momentum

The principle of conservation of momentum states that in the absence of external forces, the total momentum of a system remains constant during a collision. In two-dimensional scenarios, momentum must be conserved in each independent direction (for example, north-south and east-west axes). This principle is used to determine the post-collision velocity when objects collide and stick together, as it provides a means to calculate the resultant velocity vector from the momentum components of each colliding object.

Kinetic Energy Loss in Inelastic Collisions

In inelastic collisions, such as perfectly inelastic collisions, while momentum is conserved, kinetic energy is not. The difference between the initial and final kinetic energies is lost or transformed into other forms of energy. Understanding how to calculate the loss of kinetic energy is important for analyzing the dynamics of collisions and explaining where the energy goes, whether as heat, sound, deformation, or other dissipative processes.