Conservation of Momentum


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The conservation of momentum plays a huge role in physics. The main application of this concept is in collisions. Before we can start talking about that, however, we must first define a few terms and equations.

Important Terms


Momentum

Momentum is the product of an object mass and velocity and therefore is a vector quantity. Momentum has a unit of kilogram-meter per second. The variable for momentum is p. The equation for momentum is:

Momentum1.gif
Equation 1 The definition of momentum.

.


Elastic Collision

Collision in which both momentum and kinetic energy are conserved.

Inelastic Collision

Collision in which momentum is conserved but kinetic energy is not.

Perfectly Inelastic Collision

In a perfectly inelastic collision, momentum is conserved, kinetic energy is not, and the two objects stick together after the collision, so their final velocities are the same.

2d Inelastic Collisions

This just like a normal inelastic collision except that one or more of the objects can be moving at an angle.These types of collisions are not a major part of the AP test.

Conservation of Momentum Equation


The conservation of momentum equation is:

Momentum2.gif
Equation 2 The conservation of momentum equation
.

This equation basically says that in an isolated system (a system with no external forces), the total momentum of a system stays the same. This form of the equation is the most basic and is generally used in collisions, as we will see in the next section. The law of conservation of momentum also says that momentum cannot be created or destroyed.

Important Equations


These equations are readily used when dealing with collisions.

Momentum2.gif
Equation 2 This is the conservation of momentum equation. This equation is the first equation used when dealing with collision.


Momentum3.gif
Equation 3 This equation is used mainly in elastic collisions. This equation can be substituted into Equation 2 when there is more than one unknown.


Momentum21.JPG
Equation 4 This is the conservation of energy equation and can be used to check your answer in an elastic collision.


General Guidelines for Solving 1d Collisions


  1. Draw a picture and establish a coordinate system (Ex: Left is negative, right is positive)
  2. Solve the conservation of momentum equation for the unknown.
  3. In inelastic equations, remember that the objects stick together, so they have the same final velocity.
  4. For elastic equations, you'll have to use Equation 3 to get rid of an unknown
  5. Plug in values
  6. Don't forget the units!
  7. In an elastic equation, you can check your work by plugging in your values into Equation 4. The value on the left side of the equation (the initial energy) should equal to the value on the right side of the equation (the final energy).

Guidelines for Solving 2d Inelastic Collisions



1. Draw a picture and establish a coordinate axis. (Ex: Left is negative, right is positive. Up is positive and down is negative)
2. Break up the velocity into components as needed. (See Vector Components).
3. Write the conservation of momentum equation in the x direction and the y direction as shown in the table.
X
Momentum4.gif

Momentum13.JPG
Y
Momentum5.gif

Momentum14.JPG


4. Plug in the values you know.
5. Divide the momentum in the y direction by the momentum in the x direction.
6. Solve for the angle by taking the inverse tangent.
7. Substitute the angle back to get the final velocity.


Collision Examples



Perfectly Inelastic Collision Example

The first type of collision we will look at is a perfectly inelastic collision. This is the easiest type of collisions to work with because the problem gives you all of the values you need and there is only one unknown that you have to solve for.

An object of mass m= 20kg rolls with a velocity of 50 m/s towards another mass of m= 30 kg. What is the final velocity of the two objects given that they collide and stick together.

The first thing we need to do is draw a picture.

Momentum1.JPGMomentum2.JPG
Momentum3.JPG

The next step is to write the conservation of momentum equation. Since the two objects have the same final velocity,we van factor out the final velocity as shown.
Momentum4.JPG
Next we solve for the final velocity.
Momentum5.JPG
Now that we have the equation solved for the quantity we are looking for, we can plug in numbers and obtain the answer.
Momentum6.JPG

Elastic Collision Example

The next example we will look at is an elastic collision example.

An object of mass m= 30 kg and traveling v= 20 m/s to the right towards another object with mass m= 50 kg and velocity v= 10 m/s to the left. What will be the final velocity for each?

The first step is to draw a picture.
Momentum7.JPG


Next, we have to establish a coordinate system. Let's say that right is positive and left is negative.

The next step is to plug in values into the conservation of momentum equation and simplify. Usually you are supposed to solve the equation without any variables, but in this case it is okay to, as you will see.
Momentum8.JPG
It seems now that we are stuck because we have one equation with two unknowns. To get rid of an unknown we have to use another equation. The equation that we have to use is Equation 3. plugging in values we obtain:

Momentum9.JPG
Next we use elimination to cancel the final velocity for Object 1. (See Solving Systems of Equations)
Momentum10.JPG
Solving, we obtain
Momentum11.JPG
Now to solve for the final velocity of object 1, we have to back substitute.
Momentum12.JPG
Those are our answers and we are done with this problem. To check your work you can plug the numbers into Equation 4 and make sure you get the same answer on both sides of the equation.

2d Inelastic Collision


An object with mass 1.5 x 103 traveling east at a speed of 25.0 m/s collides at an intersection with another object with mass
2.5 x 103 kg that was traveling 20 m/s. Find the magnitude and direction of the collision, assuming that the objects stick together.

Draw a picture
Momentum15.JPGMomentum16.JPG
Plug in values into the table.

X
Momentum17.JPG
Y
Momentum18.JPG

Divide the y equation by the x equation and simplify.
Momentum19.JPG
Now back substitute into the y equation and solve for the velocity.
Momentum20.JPG
So the wreckage travels at a velocity of 15.6 m/s at an angle of 53.1 degrees north of east.

References and Additional Links


All material was adapted from Serway, Raymond A (2006), College Physics (7th ed.): Brookscole ISBN 978-0-534-99723-6 including the definitions, equations, and the example problems.

For more help on conservation of momentum, visit these sites:

Hyperphysics-Conservation of Momentum
Khan Academy
Physics Hypertextbook- Conservation of Momentum