91Ó°ÊÓ

In any collision, one of the most fundamental laws that governs the motion is the conservation of momentum. This law states that the total momentum of a closed system remains constant if no external forces act upon it. During a perfectly inelastic collision, like the one described where a car and a truck collide, momentum is conserved despite the transformation of kinetic energy into other forms like heat and sound. When we analyze momentum in this exercise, we note that before the collision, the car has a momentum of \( Mv \) and the truck has \(-2Mv \), since they are moving towards each other. Therefore, the total initial momentum is \(-Mv \). After they collide and stick together, the combined system moves with a velocity \( V_f \). Therefore, the momentum conservation equation becomes \(-Mv = 3MV_f \). Solving this helps us understand the motion characteristics post-collision.

In an inelastic collision, kinetic energy is not conserved. This is a key difference from elastic collisions where both momentum and kinetic energy are conserved. In this scenario, kinetic energy transforms into other forms of energy, like heat and sound due to the deformation of vehicles and frictional forces at play.Initially, the kinetic energy of the system is the sum of the car's kinetic energy, \( \frac{1}{2}Mv^2 \), and the truck's kinetic energy, \( \frac{1}{2}(2M)v^2 \). The sum of these energies gives us the total initial kinetic energy as \( \frac{3}{2}Mv^2 \).After the collision, as the objects stick together and move at a slower speed, their final kinetic energy is counted as \( \frac{Mv^2}{6} \). This drastically lesser final kinetic energy compared to the initial indicates a substantial transformation into other types of energy during the collision.

Momentum change in each object involved in a collision can tell us a great deal about the forces they undergo. For the car in this exercise, its initial momentum is \( Mv \), while post-collision, the car shares a common velocity with the truck, becoming \( -\frac{Mv}{3} \). This indicates a change in momentum of \( -\frac{4Mv}{3} \).For the truck, initially moving oppositely with a momentum of \(-2Mv \), the shift after collision results in a momentum of \(-\frac{2Mv}{3} \). The truck experiences a momentum change of \( \frac{4Mv}{3} \). These changes are equal in magnitude and opposite in direction, illustrating how the forces during the collision were transferred between the two vehicles.

During inelastic collisions, a big issue is the loss of kinetic energy, which is usually transformed into other forms of energy such as heat, sound, and sometimes light. This contrasts with a perfectly elastic collision where no energy is lost.In this problem, the initial kinetic energy, \( \frac{3}{2}Mv^2 \), drops significantly to \( \frac{Mv^2}{6} \) after the collision. The amount of energy lost, therefore, is \( \frac{4Mv^2}{3} \), which is a significant loss demonstrating how much energy has been converted to other forms.To find the fraction of kinetic energy lost, we take the energy lost over the initial kinetic energy and obtain a fraction of \( \frac{8}{9} \). This large fraction highlights that nearly all the kinetic energy was converted in the collision process. Understanding this helps emphasize the dramatic impact inelastic collisions have on energy distribution.