Understanding Purely Inelastic Collisions: Why Bodies Move With the Same Velocity

In the field of physics, a purely inelastic collision is a fascinating phenomenon where two bodies stick together after impact and move as a single unit. This article delves into the mechanics behind this process, focusing on the role of conservation of momentum and the unique behavior exhibited during inelastic collisions.

Conservation of Momentum in Inelastic Collisions

A purely inelastic collision is characterized by the fact that two colliding objects combine and move together after the collision. This happens due to the principle of conservation of momentum, which states that in an isolated system, the total momentum before and after the collision remains constant.

Mathematical Explanation of Momentum Conservation

To quantify this concept, we can use the following mathematical expression of momentum conservation:

$$ m_1 v_{1i} m_2 v_{2i} m_1 m_2 v_f $$

Here:

$m_1$ and $m_2$ represent the masses of the two bodies involved. $v_{1i}$ and $v_{2i}$ denote the initial velocities of the two bodies. $v_f$ is the final velocity of the combined system after the collision.

This equation expresses the fact that the total momentum before the collision is equal to the total momentum after the collision. By rearranging this equation, we can find the final velocity:

$$ v_f frac{m_1 v_{1i} m_2 v_{2i}}{m_1 m_2} $$

This calculation demonstrates that both bodies will move with the same final velocity, regardless of their individual masses or initial velocities, provided the collision is purely inelastic and the bodies stick together.

Nature of the Collision: Kinetic Energy and Momentum

It's important to note that while momentum is always conserved in a collision, kinetic energy is not. In the case of a purely inelastic collision, some kinetic energy is converted into other forms, such as heat or sound. However, the conservation of momentum ensures that the final velocity of the combined mass is determined by the initial velocities and masses of the colliding objects.

Summary of Key Concepts

In summary, bodies move with the same velocity after a purely inelastic collision because they combine into a single system. The conservation of momentum dictates that their final velocity is the weighted average of their initial velocities, resulting in a common velocity. This phenomenon is a cornerstone of classical mechanics and plays a crucial role in understanding the behavior of objects during inelastic collisions.

Frequently Asked Questions

1. Why does momentum remain constant in inelastic collisions?
Momentum conservation is a fundamental principle in physics, and it applies to all collisions, whether elastic or inelastic. In an inelastic collision, the total momentum remains constant because no external forces act on the system to change it. The conservation is primarily visible through microscopic particle interactions, which manifest as changes in temperature.

2. What happens in a perfectly inelastic collision?
In a perfectly inelastic collision, the two bodies stick together and move as a single unit. The definition of a perfectly inelastic collision is that the objects combine to form a single object after impact, resulting in the same velocity for both.

3. Why do bodies move together with the same velocity in a purely inelastic collision?
The bodies move together with the same velocity because of the conservation of momentum. The combined mass of the bodies after the collision weighs the initial velocities in such a way that a common velocity is achieved. This is a result of the colliding bodies becoming a single system, sharing the same velocity post-collision.