Conservation of Linear Momentum in Inelastic Collisions

In an inelastic collision, two particles collide and stick together, resulting in a loss of kinetic energy. During this type of collision, the total linear momentum is conserved, but the total kinetic energy is not conserved.

Definition of Linear Momentum

Linear momentum is a vector quantity that depends on the mass and velocity of an object. Mathematically, it is given by the product of mass and velocity:

Momentum (p) = mass (m) * velocity (v)

Conservation of Linear Momentum

According to Newton's third law of motion, the total linear momentum of an isolated system remains constant before and after the collision, provided no external forces act on the system. This principle is known as the conservation of linear momentum.

In an inelastic collision, the two particles collide and stick together, forming a single object. Since the particles stick together, they move with the same final velocity after the collision. As a result, the total momentum of the system before and after the collision remains the same.

Mathematical Representation

Let's consider two particles with masses m1 and m2, initially moving with velocities u1 and u2, respectively. After the collision, they stick together and move with a final velocity v. The conservation of linear momentum can be expressed as follows:

Initial momentum = Final momentum
(m1 * u1) + (m2 * u2) = (m1 + m2) * v

Conservation of Kinetic Energy

In an inelastic collision, the total kinetic energy of the system before the collision is not equal to the total kinetic energy after the collision. Some kinetic energy is lost during the collision, typically transforming into other forms of energy such as heat, sound, or deformation.

This loss of kinetic energy is primarily due to the fact that the two particles stick together instead of bouncing off each other. The energy required to permanently deform the particles or overcome intermolecular forces results in a decrease in kinetic energy.

Conclusion

In summary, during an inelastic collision of two particles, the total linear momentum is conserved, but the total kinetic energy is not conserved. This is because the particles stick together and move with a common final velocity, resulting in a loss of kinetic energy.