Chapter 6 - Linear Momentum and Collisions - Learning Path Questions and Exercises - Conceptual Questions - Page 214: 13

In an inelastic collision, some of the kinetic energy of the colliding objects is converted into other forms of energy, such as heat or sound, or is used to deform or break the objects involved in the collision. Even though kinetic energy is lost, the total momentum of the system is still conserved. This can be explained using the principle of conservation of momentum, which states that the total momentum of a closed system is conserved when there are no external forces acting on it. In an inelastic collision, the colliding objects stick together or deform, which means that their velocities change. However, the total momentum of the system before and after the collision is the same because there are no external forces acting on the system. This means that the sum of the momenta of the objects before the collision is equal to the sum of the momenta of the objects after the collision. The conservation of momentum alone does not determine what happens to the kinetic energy of the system. Kinetic energy is not conserved in an inelastic collision because some of it is converted into other forms of energy. The amount of kinetic energy lost depends on the nature of the collision and the objects involved. In general, the more elastic a collision is, the less kinetic energy is lost. In a perfectly elastic collision, no kinetic energy is lost, and the objects bounce off each other with the same velocity they had before the collision. So, even though some of the kinetic energy is lost in an inelastic collision, the conservation of momentum still holds true because there are no external forces acting on the system, and the total momentum remains constant.

Conservation of Energy $ E_{i} = E_{f}$ $ \frac{p_{i}^{2}}{2m} = \frac{p_{f}^{2}}{2(m+m)} $ Since $ p_{i} = mv_{1i} $ and $p_{f} = (m+m)v_{f} $ And we know from the Conservation of momentum $p_{i} = p_{f}$ $mv_{1i} = (m+m)v_{f}$ $v_{f} = \frac{mv_{1i}}{m+m}$ Pluging this into the Conversation of energy equation we see that the statement is false.