1.5 Elastic and Inelastic Collisions

1.5 Elastic and Inelastic Collisions

Elastic Collisions:

In an elastic collision, both momentum and kinetic energy are conserved. The objects involved in the collision do not undergo any permanent deformation, and no energy is lost to heat or sound.

Example:

Two billiard balls colliding on a pool table, where both balls bounce off each other with no loss of kinetic energy.

Inelastic Collisions:

In an inelastic collision, momentum is conserved, but kinetic energy is not. Some of the kinetic energy is transformed into other forms of energy, such as heat or sound, and the objects may stick together or deform.

Example:

A car crash, where the vehicles may crumple and stick together, and some of the kinetic energy is converted into heat and sound.

Example Problem:

Two cars collide inelastically. Car A has a mass of 1000 kg and is moving at 20 m/s, while Car B has a mass of 1500 kg and is moving at 10 m/s in the opposite direction. If they stick together after the collision, what is their final velocity?

Solution:

Using the conservation of momentum:
m_A * v_A + m_B * v_B = (m_A + m_B) * v_final
Substitute the known values:
(1000 kg) * (20 m/s) + (1500 kg) * (-10 m/s) = (1000 kg + 1500 kg) * v_final
Solve for v_final:
v_final = (20,000 kg·m/s - 15,000 kg·m/s) / 2500 kg = 5000 kg·m/s / 2500 kg = 2 m/s
The final velocity of the combined cars is 2 m/s in the direction of Car A's initial motion.

Final Solution:

(1000 kg) × (20 m/s) + (1500 kg) × (-10 m/s) = (1000 kg + 1500 kg) × v_final
v_final = (20,000 kg·m/s - 15,000 kg·m/s) / 2500 kg
v_final = 2 m/s
The final velocity of the combined cars is 2 m/s in the direction of Car A's initial motion.