Under construction! Site being updated for 2024-25 AP Physics changes

4-Momentum

Slideshows:
Momentum & Impulse Slideshow (previous version);
Unit 4 Impulse & Momentum Slideshow (under construction, but updated for 2024-25)

Textbook: Chapter 5 & chapter 6.7 in Mastering Physics (get online code for registration on about page of google classroom)

What's in this unit?

This unit explores the relationships between force, time, impulse, and linear momentum via calculations, data analysis, designing experiments, and making predictions. We learn ways to express and use the law of conservation of linear momentum of objects and systems, especially for collisions based on Newton’s third law. We make connections between momentum and kinetic energy of objects or systems and see under what conditions these quantities remain constant.

Topic 4.1 Linear Momentum

Collision: an interaction where the forces between objects are much greater than the net external forces exerted on the objects.

Explosion: an interaction where the internal forces to a system cause the objects in the system to move apart.

Linear momentum: equal to mass x velocity. Units are kg m/s.

a vector quantity that has the same direction as the velocity
used to analyze collisions and explosions
linear momentum is similar to, but different from angular momentum. If the term momentum is used, it tends to refer to linear momentum.

Topic 4.2 Change in Momentum & Impulse

Change in momentum: Δp = pf - pi = mvf - mvi

Impulse: the product of force x time for the average external force exerted on a system (J = FΔt) measured in N s or kg m/s.

Impulse-momentum theorem: the impulse acting on a system is equal to the change in momentum of the system. (J=FΔt=Δp)

Newton's 2nd law is actually an extension of impulse-momentum theorem (see the equation showing how Fnet=ma has the impulse momentum between the parts).

Frequent tasks:

Find the impulse acting on an object as the area under a F(t) graph (based on J = FΔt)
and extend to find the change in velocity (based on Δp = mΔv).
Find the average force acting on an object as the slope of a p (t) graph (based on F = Δp/Δt)

Topic 4.3 Conservation of Linear Momentum

The velocity of the center of mass of a system can be found by the sum of the momenta of the objects over the sum of their masses.

Momentum is conserved in all interactions.

If the net external force on a system is zero, the total momentum of the system is constant
- any change in momentum for one part of the system must be balanced by an equivalent and opposite change of momentum elsewhere in the system;
  - this is an application of Newton's 3rd law because the impulse on one object in a collision or explosion must be equal and opposite to the impulse on the second object
- Equations are build upon the principle that in the absence of an external force there is no change in momentum. Σpinitial =Σpf
  - In a 2 object system this is most often becomes m1v1-initial + m2v2-initial = m1v1-final + m2v2-final
  - students will be able to manipulate this and adapt it to a variety of situations and to support claims.
- In a 2D collision, the conservation of momentum is applied separately to X and Y axes.
If the net external force on a system is nonzero, momentum is transferred between the system and the environment
- The impulse-momentum theorem describes that the change in momentum of the system's center of mass based on the net force and time

Topic 4.4 Elastic and Inelastic Collision

Elastic Collision: a collision in which the total initial kinetic energy and total final kinetic energy of the system are equal

Inelastic Collision: a collision in which some kinetic energy is transformed by nonconservative forces into other forms of energy

a perfectly inelastic collision is one where the objects stick together and move with the same velocity (as each other) after the collision.
m1v1-initial + m2v2-initial = (m1+ m2)vfinal

Don't be fooled: collisions where objects bounce apart can be inelastic or elastic. The way to tell them apart is to do the math to see if the kinetic energy is conserved.

it is possible (but not consistent with how this course has been tested) to derive an equation from the conservation of energy and the conservation of mass that shows that the relative velocity between objects does not change during an elastic collision. This shortcut can be used to help differentiate these two collision types.

Videos

Watch bullet fly right Through thick aluminum

Testing the limits of a golf ball

very cool way to see what happens to the energy and momentum of an object during a collision sprinkled throughout these exciting high speed tests.

Common Misconceptions

Misconception: Momentum is not a vector.
- Principle: Momentum really is a vector. We can tell because a collision where two objects with the same momentum magnitude by opposite directions will result in a zero speed after colliding. This is only explained by have a sum of zero momentum, which would be true if momentum is a vector. This also explains why a an object of lower mass bounces back off a stationary object of larger mass after a collision.
Misconception: Conservation of momentum applies only to collisions.
- Principle: Momentum is conserved in all closed systems or can be tracked as impulse in an open system. Challenge: Apply conservation of momentum to these non collisions: the explosion of a firecracker or a person in space throwing another object (or even just continuing motion).
Misconception: Momentum is the same as force.
- Principle: Changes in momentum are caused by force (impulse momentum theory). As a result faster moving objects or larger objects are able to deliver a greater force than smaller or slower objects when colliding, but an object can have momentum without interacting with other objects and therefore momentum cannot be the same as force.
Misconception: Moving masses in the absence of gravity do not have momentum.
- Principle: momentum is a property intrinsic to an object with mass and velocity. I really don't get why anyone would think this, but space is an excellent scenario for conservation of momentum because it can so often be considered as a closed system. So, when you see the center of mass has a velocity of v and there is separation of the system, you can know that the center of mass of the system maintains that same velocity v even though the individual objects have different speeds.
Misconception: The center of mass of an object must be inside the object.
- Principle: The center of mass of an object can be found by taking the average position (X, Y, Z coordinate) of all the particles in an object. This is a mathematical value with importance in how things behave, but without physical substance. Consider a donut or a two cart system.
Misconception: Center of mass is always the same location as the center of gravity.
- This detail is not relevant for our course, but I hate to delete things. Center of mass is the average location of the mass of all the particles in an object whereas center of gravity is the average location of force of gravity on all the particles. Really tall objects experience a slightly different g value at the top of their structure than at the bottom, so the particles at the top have a slightly smaller weight than similar particles at the bottom, causing the center of gravity to be slightly lower than the center of mass. This actually matters in some large scale engineering applications, but would not be tested in AP Physics.
Misconception: Momentum is not conserved in collisions with "immovable" objects
- Principle: Newton's 2nd law: the immovable object is simply experiencing a much smaller acceleration (a=F/m) because it is either much more massive or attached to an object more massive that whatever you have colliding with it.
- 2nd Principle: How you define your system determines if the system is open or closed. If you define your system as the object that is changing momentum you will never have momentum conserved within your system, but if you include all the interacting objects in your system you will have momentum conserved. So, the immovable object is only immovable because it is attached to the earth (like a wall), and if you include the earth in your system then an object that experiences a ∆p in the +x direction will have had the earth experience a ∆p in the -x direction as a result. Of course, given the mass of the earth, there is no noticeable result of this, just like jumping off the earth causes additional force on the earth, but we don't notice any change in motion of the earth from that either.
Misconception: Momentum and kinetic energy are the same.
- Principles: momentum is directly related to velocity; kinetic energy is quadratically related to velocity; mechanical energy is not conserved in any real collisions (only ideal elastic collisions). When collisions happen people want to talk about transferring energy and momentum interchangeably. In isolated collisions, though, mechanical energy is frequently not conserved, which is how we even prove that momentum is a thing. Take the case of a collision where a 1 kg object going 1 m/s runs into, and sticks to a 1 kg stationary object. Together they now have a mass of 2 kg, conservation of momentum predicts that 1(1) + 0 = (1+1)v, the final speed will be 0.5 m/s. If you consider energy, you have 1/2 (1)(12)+ 0 = 0.5 J before the collision and 1/2 (2) (0.52) = 0.25 J after, so kinetic energy was not conserved (some was converted to non-mechanical forms of energy). Yes energy was transferred and momentum was transferred, but they were not transferred in the same way and are not the same thing.

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