4.3 Conservation of Linear Momentum and Collisions

Conservation of Linear Momentum in a System🎱

Conservation of Linear Momentum

** The total momentum, magnitude, and direction are conserved in a system unless acted on by an outside force. **

Here are some key things to remember about the conservation of linear momentum:

• Conservation of linear momentum states that the total linear momentum of a closed system remains constant unless acted upon by an external force.
• Linear momentum is a vector quantity, meaning it has both magnitude and direction.
• The linear momentum of an object is equal to its mass multiplied by its velocity, represented mathematically as p = m*v.
• The total linear momentum of a system of objects is equal to the vector sum of the linear momenta of all the individual objects in the system.
• In an elastic collision, the total linear momentum before and after the collision must be the same, assuming no external forces are acting on the system.
• In an inelastic collision, the total linear momentum is still conserved but kinetic energy is not.
• Conservation of linear momentum can be used to analyze the motion of objects in a variety of situations, such as collisions, explosions, and the motion of objects in a resistive medium.
• In the context of rocket propulsion, the conservation of linear momentum can be used to analyze the motion of the rocket and the exhaust gases it produces.
• The conservation of linear momentum is a fundamental principle in physics and is closely related to Newton's laws of motion.
• It is also closely related with the principle of action and reaction which states that for every action there's an equal and opposite reaction.

This law of conservation can also be written as:

The initial momentum is equal to the final momentum, if there was no impulse. 

⚠️So...why is momentum conserved in a collision?

We have to blame it on Newton, again! Think back to Newton's Third Law: every action has an equal and opposite reaction. This means that there is an equal and opposite force acting on the two objects that collided.

So let's see if there's an impulse from a collision. Since impulse is equal to the change in momentum, if it is 0, we know that momentum was conserved. The change in momentum of both objects is the same magnitude, but different directions. So the sum of these is zero!

Collisions

A collision is the most basic application of conservation of momentum, from pool balls knocking each other to a rocket launching, collisions occur daily. They typically involve a system of masses interacting with some speed. The forces these masses enact on each other do not affect the conservation of momentum of the system, because there is no net force!

Below is an example of a scenario before and after, where momentum is conserved:

Taken from Lumen Learning

Elastic Collisions:

• Momentum is conserved

• Mechanical energy is conserved

• Kinetic energy is conserved

• Sometimes referred to as "Bouncy" collisions Inelastic Collisions:

• Momentum is conserved

• Mechanical energy is conserved

• Kinetic energy is not conserved

• If a collision is perfectly inelastic or "sticky", the max amount of kinetic energy is lost Here's a fun interactive cart lab you can mess around with! Check out the different types of collisions and how velocity and mass affect the system. 

Practice Questions

Taken from College Board

(For part 1 you'll need the answer for part a of this question, which is -2.03 N*s)

Answer: Impulse is equal to the change in momentum, and in this scenario mass is not changing so we can look at this problem with initial and final velocity. Remember that velocity is a vector, so positives and negatives represent direction!

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Answer:

Use conservation of momentum to calculate the speed of cart 2. Then, you should recall that kinetic energy is conserved in elastic collisions.