# PHY C4: Momentum

Today we’ll cover:

• Linear momentum
• Principle of Conservation of Momentum
• Collisions
• Kinetic energy in collisions

Let’s go!

Total Momentum
Newton’s 1st Law states that when no external force is acting on an object, its momentum does NOT change.
This applies to systems as well.
A system with two bodies (e.g. 2 balls floating in space) will not experience a change in NET momentum if there is no external force.

Let’s say both of these balls were in motion towards each other.
They each have a different momentum initially.

Once they collide, they will both experience a CHANGE in momentum.
Both balls exert a force on each other.
However, there is NO external force acting on the system as a whole.
Thus, the NET momentum of the system CANNOT change.

To calculate total momentum of a system, consider the respective momenta of each object as a separate vector.
Use VECTOR ADDITION to add all the momenta together.
This will give you the TOTAL momentum of a system.

Principle of Conservation of Momentum
“If no external force acts on a system, the total momentum of the system is conserved.”

pinitial = pfinal

Since p = mv,
m1u1 + m2u2 = m1v1 + m2v2

What are Collisions?
A type of system where 2 or more objects collide with each other.
Before the collision, each object may have a separate momentum.
After the collision, each object’s momentum may change.
However, the NET momentum is ALWAYS conserved.

Collisions can be considered as 1-dimensional or 2-dimensional.

One-Dimensional Collisions
Collisions where all objects travel along the same axis.
In this case, the momenta are all linear, & can be dealt with algebraically.
No vector addition needed.

Two-Dimensional Collisions
Sometimes, your objects will NOT be travelling linearly.
To solve these problems, you need to apply RESULTANT VECTORS.

REMEMBER that the vectors represent the MOMENTUM (mass x velocity) of each object, NOT the velocity.

 STEP 1 STEP 2 STEP 3 Find the total momentum BEFORE the collision Since pinitial = pfinal , the total momentum AFTER the collision is the SAME as before Given the final momentum of 1 object, you can find the momentum of the other: pf1 + pf2 = pfinal

To find TOTAL momentum, there are 2 methods you can use:

1. Using Vector triangles
2. Using Resolution of vector components
• When using Vector Triangles, the 2 initial momenta of the objects are the 2 sides of the triangle. The 3rd side is the resultant (TOTAL) momentum.
• When using Resolution, the 2 perpendicular components are referred to as
“Northerly” & “Easterly” Components
OR
“Vertical” & “Horizontal” Component
(depending on where you are looking at).

To find values such as MASS or VELOCITY of one of the objects, just remember that p = mv.
If you can find the momentum of an object, you can find mass or velocity as long as you’re given the other value.

Remember: the direction of VELOCITY is the same as the direction of MOMENTUM of an object.
Finding the direction of momentum can give you the direction of velocity.

Now that we have seen the methods to calculate momentum & velocity in a collision, let’s explore the TYPES of collision.
There are 2 types of collisions: Elastic & Inelastic.

What are Elastic Collisions?
Collisions where TOTAL KINETIC ENERGY is CONSERVED.
In an elastic collision, the total kinetic energy of a system before the collision & after are EQUAL.

Before the collision, the objects possess kinetic energy.
During the collision, some kinetic energy is stored as potential energy in the objects.
If the collision is perfectly elastic, all of this potential energy is converted back into kinetic energy.

This means that there is NO loss of energy via heat, sound, or deformation.

Total kinetic energy before = total kinetic energy after
½m1u12 +  ½m2u22 =  ½m1v12 +  ½m2v22

Based on this equation, you can prove* another fact:
u1 – u2 = v1 – v2

This shows that the RELATIVE VELOCITY of APPROACH is EQUAL to the RELATIVE VELOCITY of SEPARATION.

In other words, the relative velocity between two objects before a collision is the same as the relative velocity after the collision.

*Here’s the proof (from Redefining Knowledge):

What are Inelastic Collisions?
Collisions where total kinetic energy is NOT conserved.
Some energy is transformed into sound, heat or permanent deformation of the object.
All of the above equations do NOT apply here.
However, the principle of conservation of MOMENTUM still applies.

In reality, most macroscopic collisions are inelastic.
The only TRULY perfect elastic collisions occur on a MICROSCOPIC scale: atoms & particles colliding with each other.

Here’s a summary of the characteristics of each collision:

 Aspect Elastic Inelastic Total Momentum Conserved m1u1 + m2u2 = m1v1 + m2v2 Conserved m1u1 + m2u2 = m1v1 + m2v2 Total Kinetic Energy Conserved ½m1u12 +  ½m2u22 = ½m1v12 +  ½m2v22 Not conserved ½m1u12 +  ½m2u22 ≠ ½m1v12 +  ½m2v22 Relative speed Conserved u1 – u2 = v1 – v2 Not conserved u1 – u2 ≠ v1 – v2

Lastly, there’s one more type of event where we can apply the principle of conservation of momentum: EXPLOSIONS.

What is an Explosion?
A situation where two or more objects initially at rest suddenly gain momentum.

E.g. the firing of a gun:
The bullet & the gun are initially at rest, but after the explosion, both gain momentum in opposite directions – the bullet is shot, & the gun recoils.

Although each object gains momentum, the NET momentum must ALWAYS be conserved: it remains 0.
This is because all the objects move in opposing directions.
If you calculate each momentum as a vector, & add all the vectors together,
the sum MUST be 0.

sum of initial momentum = sum of final momentum
0 = sum of initial momentum
0 = m1v1 + m2v2