Momentum and Collisions

1. Define momentum 2 ways.

   1. How hard it is to stop an object
   2. P ≡ mv
2. State Newton's second law in terms of momentum. S F = dP / dt
3. Name two types of situations where momentum is conserved.
   1. Explosions - things flying apart
   2. Collisions - things flying together
4. State one of the key reasons momentum can be conserved. - It's a vector!
5. Solve 1 dimensional "explosion" problems using conservation of momentum.
6. Describe the difference between elastic and inelastic collisions.

Collision Type	Conservation  Momentum	Conservation Energy	Conservation Kinetic En.	Particles Stick
Elastic	Yes	Yes	Yes	No
Inelastic	Yes	Yes	No	Yes

7. Solve 1 dimensional inelastic collision problems using conservation of momentum.

Homefun: Read 9.1 & 9.3, Questions 1-10 p. 281; Problems 1, 3, 5, 19.

Homefun: Questions 1 - 11 p.44-45; prob 1, 3, 5, 7, 9

Read: Insultingly Stupid Movie Physics

Chapter 7, Movie Momentum: The Attractive Force of Glass, Rail-Gun Recoil, and Cosmic Toyotas, pp 99 - 115

Metacognition Problem Solving Question 9.1: Do objects collide or fly apart in the problem? If they do then the problem can often be solved using conservation of momentum. Solutions for final velocity of an object in one dimensional momentum problems usually end up with a mass ratio time the starting velocity.

Lesson 1

Key Concept: Conservation of momentum

Purpose: Understand how to use conservation of momentum in problem solving.

Interactive Discussion: Objective 1 - 4. Note that momentum is one of the great conservation laws of physics.

Demo 9.1:  Objective 3b. Explain that the spheres-on-a-rod will exhibit conservation of momentum when dropped. On the way down the total mass is the combined mass of the spheres. On the way up, only the small sphere will rise. Guess what happens to its velocity?

Video Clip: Show a video clip of the rail gun scene in Eraser.

In Class Problem Solving: Objective 5 

1. Arnie fires the rail gun.
2. Bubba pushes Nancy.

Demo 9.2:  Objective 6. Demonstrate the difference between elastic and inelastic collisions using happy and sad spheres.

In Class Problem Solving: Objective 5 

1. A moving train car collides with a stationary car.
2. Two meteors with different velocities collide and fuse.

Resources/Materials: spheres-on-a-rod, happy and sad spheres

Elastic Collisions

Relevance: Elastic collisions are useful for analyzing numerous problems from gas dynamics (collisions between molecules) to billiards.

8. Describe in qualitative terms what happens in one dimensional elastic collisions. 

9. Solve 1 dimensional elastic collision problems.

10. Solve problems inelastic collisions in 2 dimensions.

11. Solve problems involving elastic collisions in 2 dimensions.

When a mass collides with a stationary mass of the same size in a glancing elastic collision, the velocity vectors form a 90 degree angle afterwards.(p.270)

Metacognition Problem Solving Question 9.2: Can I break the vectors into components? The most complex vector problem can always be broken up into simpler 1 dimensional problems. Hence, The sum of  momentums in each dimension after a collision has to be the same as the sum of momentums in the same dimension before the collision. - Think components!

Homefun: prob.15, 19, 21, 33, 37

Lesson 2

Key Concept: Conservation of Momentum With Elastic Collisions and in Two Dimensions.

Purpose: Expand the understanding of how conservation of momentum can be used for problem solving.

Demo 9.3:  Objective 9.  Demonstrate elastic collisions on an air track with combinations of various sized cars.

In Class Problem Solving: Objectives 9

1. Derive an expression for the final velocities after the elastic collision of equal mass objects in one dimension.

Interactive Discussion: Objective 10 - 11.

In Class Problem Solving: Objectives 10 -11

1. When Bob meets Jane at an intersection
2. Bouncing billiard balls go boink.

Resources/Materials: Air track.

Impulse Section 9.2

12. Solve combined conservation of energy and momentum problems for inelastic collisions.

13. Solve combined conservation of energy and momentum problems for collisions involving springs.

14. Define impulse mathematically 2 different ways and explain in general terms what it indicates.

• (imp) = ∫ F_(t) ∙ dt
• (imp) = ΔP  
• Roughly speaking: an indication of how much effect a force will have on changing an object's momentum.

Note: impulse is a vector.

 

Metacognition Problem Solving Principle 9.3: Remember that impulse obeys Newton's third law! In other words, during a collision, the change in momentum caused by the impulse one object exerts on a second is exactly the same as the change in momentum caused by the impulse the second object exerts on the first.

Homefun: Read 9.2, prob. 6, 27, 57, 61 

Lesson 3

Key Concept: Impulse

Purpose: Understand what happens during a collision.

Interactive Discussion: Objective 12. How much force can I take?

In Class Problem Solving: Objectives 12

1. Ballistic pendulum.

2. Bob and Jane play chicken with bumper cars (collisions with springs).

3. Bozo burns the rope on the bozomobile (releasing springs).

Interactive Discussion: Objective 13 . How much force can I take?

Demo 9.4:  Objective 12, 13.  Demonstrate the effects of elastic and inelastic collisions on a block of wood using happy and sad sphere pendulums. Which has the higher impulse?

In Class Problem Solving: Objectives 9

Resources/Materials: Happy and sad sphere pendulums.

Center of Mass Section 9.6

15. Define center of mass

Discrete objects	Xcm = ∑ Xi mi ∑ mi	Ycm = ∑ Yi mi ∑ mi
General case	Xcm = 1/m ∫ X ∙ dm	Ycm = 1/m ∫ Y∙ dm
English	the average spatial location of an object's mass

16. Find the center of mass of various objects.

17. State whether an object will rotate, translate, or both by looking at the location of forces relative to the center of mass.

18. Explain how center of mass relates to the  stability of an object.

19. State what happens to the center of mass of an object if it exploded in outer space. 

Metacognition Problem Solving Principle 9.4: After calculating the center of mass coordinates for an object, does it look like the object could be balanced by placing a thin support under the center of mass? Center of mass is like a balance point. By thinking of it in this manner, it's often possible to judge if a center of mass calculation makes sense by asking if the object would balance if suspended from

Homefun: Read 9.4 thru 9.6, prob. 39, 41 

Lesson 4

Key Concept: Center of mass

Purpose:  Calculate the center of mass and use it as a problem solving concept in momentum problems.

Interactive Discussion: Objective 15. All of an object's mass can be considered to exist at it's center. How can the center of mass be found experimentally?

Demo 9.4:  Objective 16. Throwing a meter stick. Throw a meter stick so that it rotates and so that it translates. What is the difference in how the force is applied?

Demo 9.5:  Objective 16. Attempting to stand on one leg. Stand with the feet spread apart shoulder width. Attempt to raise one foot and stand on only one foot without changing the position of your center of mass. (It can't be done)

Demo 9.5:  Objective 16. Throwing an opponent who attempts to punch you (Aikido). Have a subject stand on one foot with the back leg extended backwards and extend his fist as far forward possible. This simulates the act of rushing forward while throwing punch. Have a second person very lightly pull forward on the wrist of the first person's extended arm. (This will very easily unbalance the  first person. If the first person were actually rushing forward while throwing a powerful punch, he would be violently thrown to the ground with both translational and rotational motion)

In Class Problem Solving: Objectives

Rockets and Explosions Section 9.6

Relevance: Want to learn rocket science? Here it is. Conservation of momentum is the basis principle that makes rockets go.

20. State what happens to the center of mass of an object when the parts of an object are pulled together by an internal force (assuming no external resistance forces). 

21. State what happens to the center of mass of a rocket (including its fuel) when it is accelerating in outer space.

22. Calculate the thrust produced by a rocket.

    F_T = v_m dm/dt

     

23. State two reasons why a rocket must have a very large supply of fuel in order to move from the surface of Earth into outer space.

Metacognition Problem Solving Principle 9.4: Remember that the location of an object's center of mass of a system is only affected by external forces. Any form of explosion in an unrestrained object will generate internal forces.

Homefun: Read section 9.7, prob. 43, 51 

Lesson 5

Key Concept: Propulsion using momentum and center of mass 

Purpose:  Understand how rockets work.

Interactive Discussion: Objective 15. 

Demo 9.4:  Robot which propels itself by changing the location of its center of mass.

In Class Problem Solving:

Tsiolkovsky's rocket equation, or ideal rocket equation is named after Konstantin Tsiolkovsky who independently derived it and published in his 1903 work,^[

where:

m₀ is the initial total mass, including propellant, in kg (or lb)

m₁ is the final total mass in kg (or lb)

v_e is the effective exhaust velocity in m/s or (ft/s) 

is the delta-v in m/s (or ft/s)

Some typical values of the exhaust gas velocity V_e for rocket engines burning various propellants are:

• 1.7 to 2.9 km/s (3800 to 6500 mi/h) for liquid monopropellants
• 2.9 to 4.5 km/s (6500 to 10100 mi/h) for liquid bipropellants
• 2.1 to 3.2 km/s (4700 to 7200 mi/h) for solid propellants

Formal Lab Investigation

Level the air track so that the sliders remain stationary unless they are pushed.

Place 2 sliders of equal mass on the track. Just before the collision, the one in back will be sliding toward the stationary one in front.

Set up a pair of photogates so that the first measures the moving slider's velocity just before the collision and the second measures the velocity of the 2nd cart just after the collision.

Gently push the slider in back, measure the results of the collision, and determine if conservation of momentum is a good model for the system.