Collisions: Conservation of Momentum and Types of Collisions

Collisions

Introduction to Collisions

Collisions are interactions between two or more bodies where they exert forces on each other for a short period. The study of collisions is fundamental in physics, as it helps explain the conservation laws and the behavior of objects in motion.

• Collision: An event where two or more objects come into contact and exchange energy and momentum.

• Types of collisions: Elastic and inelastic collisions.

Conservation of Momentum

Principle of Conservation of Momentum

Momentum is a conserved quantity in physics, meaning the total momentum of a closed system remains constant if no external forces act on it. This principle applies to all types of collisions.

• Momentum: Defined as the product of an object's mass and velocity ($p = mv$).

• Conservation Law: The total momentum before a collision equals the total momentum after the collision.

Equation:

$M_1V_{B1} + M_2V_{B2} = M_1V_{A1} + M_2V_{A2}$

• System: Must consider all possible sources and objects in the system for conservation.

Conservation Practice Example

Consider a boater initially at rest in a static boat. If the boater steps out of the boat and onto a dock with a velocity of 2.5 m/s to the right, what is the velocity of the boat?

• Application: Use conservation of momentum to solve for the boat's velocity after the boater steps out.

• Example: If the total initial momentum is zero, the momentum of the boater and boat after stepping out must sum to zero.

Elastic Collisions

Definition and Properties

Elastic collisions are those in which both momentum and kinetic energy are conserved. The colliding objects retain their original shapes and move separately after the collision.

• Elastic Collision: A collision where both momentum and kinetic energy are conserved.

• Objects: Do not deform permanently and move apart after the collision.

Equations:

• Conservation of momentum: $M_1V_{B1} + M_2V_{B2} = M_1V_{A1} + M_2V_{A2}$

• Conservation of kinetic energy: $\frac{1}{2}M_1V_{B1}^2 + \frac{1}{2}M_2V_{B2}^2 = \frac{1}{2}M_1V_{A1}^2 + \frac{1}{2}M_2V_{A2}^2$

Elastic Practice Example

A 0.025 kg marble moving to the right at 0.200 m/s undergoes an elastic head-on collision with a 0.030 kg marble moving to the left at 0.180 m/s. After the collision, the smaller marble moves left at 0.315 m/s. What is the velocity of the larger marble after the collision? Assume the surface is frictionless.

• Application: Use conservation of momentum and kinetic energy to solve for the unknown velocity.

• Example: Set up equations for both conservation laws and solve simultaneously.

Inelastic Collisions

Definition and Properties

Inelastic collisions are the opposite of elastic collisions. In these collisions, two objects collide and result in a loss of kinetic energy, often sticking together and moving as a single mass. Perfectly inelastic collisions are a special case where the objects stick together after the collision.

• Inelastic Collision: A collision where momentum is conserved but kinetic energy is not.

• Perfectly Inelastic Collision: The colliding objects stick together and move as one mass after the collision.

Equation for Perfectly Inelastic Collision:

$M_1V_{B1} + M_2V_{B2} = (M_1 + M_2)V_A$

• General Inelastic Collision: Objects may deform and move separately after the collision, but do not stick together.

Inelastic Practice Example

A 1500 kg luxury sedan stopped at a traffic light is hit from behind by a convertible with a mass of 975 kg. The two cars become entangled. If the convertible was moving at a velocity of 22.0 m/s before the collision, what is the velocity of the cars after the collision?

• Application: Use the equation for perfectly inelastic collisions to solve for the final velocity.

• Example: Substitute the given masses and velocities into the equation and solve for $V_A$.

Summary Table: Types of Collisions

Type of Collision	Momentum Conserved?	Kinetic Energy Conserved?	Objects After Collision
Elastic	Yes	Yes	Move separately, no permanent deformation
Inelastic	Yes	No	May stick together (perfectly inelastic) or move separately with deformation

Homework

Physics Unit 14 Assessment

Review the concepts of momentum conservation, elastic and inelastic collisions, and practice solving problems involving these principles.

• Recommended: Attempt practice problems and review the equations for each type of collision.

Additional info: The cartoon about science teachers and playground duty humorously references the application of physics concepts (momentum and collisions) in everyday life.