Unit 6: Momentum

1. Short Unit • No Quiz • Test Unit 6: Momentum • Momentum • Impulse • Conservation of Momentum

2. Introduction 1500 m/s 10 m/s .02 kg bullet 10 kg bowling ball Which of these objects would be the hardest to stop? 3 m/s 3×108 m/s 80 kg running back 2×10-27kg neutron

3. Momentum • Inertia: How hard it is to change an object’s motion (velocity) • But we must also consider how much you want to change the velocity ! • Momentum – a measure of how hard it is to stop a moving object.

4. Momentum Calculating momentum: Momentum = mass × velocity Equation: p = mv (Why p? Well, “m” was already taken!) • Units of momentum are kg·m s • It’s a vector! Direction is the same as velocity • Note that when an object is standing still (v=0), it has no momentum.

5. 1500 m/s 10m/s .02 kg bullet 10 kg bowling ball Which of these objects would be the hardest to stop? p=mv = 30 kg·m/s p=mv = 100 kg·m/s 3 m/s 3×108 m/s 80 kg running back 2×10-27kg neutron p=mv = 240 kg·m/s p=mv = 6x10-19 kg·m/s

6. Unit 6: Momentum • Momentum • Impulse • Conservation of Momentum

7. Changing Momentum • Momentum tells us how hard it is to stop a moving object • It’s useful to find out how much force is required to change an object’s momentum

8. Newton’s 2nd Law: If I apply a force continuously, it accelerates continuously It accelerates while the force is applied Then stops accelerating, but with a new velocity- so the momentum has changed Temporary Force What if I apply a force for just a little while?

9. Force x the time the force is applied Change in Momentum Impulse Impulse (J) – when a force is exerted over a limited period of time, causing a change in velocity, and therefore a change in momentum. J= F t = Δp Each side is equal to the impulse

10. Change of Momentum So F t = Δp Can be written as F t = mΔv Note: since p = mv, and mass is usually constant, Δp = m Δv mΔv is not on your equation sheet– you’ll need to remember it!

11. Impulse Example Practice Problem: If you exert a 300N force on a .08 kg golf ball for .04 seconds, how fast will the golf ball fly? F t = Δp F t = m Δv (300N)(.04s) = (.08kg) Δv Δv = 150 m/s

12. Impulse Example A speeding train is headed for a cliff, and the tracks are out!! The train lost its brakes, but luckily, Superman is nearby. If the train is moving at 24 m/s and has a mass of 10,000 kg, how long will it take Superman to stop the train if he can exert 12,000 N on it? F t = m Δv (12,000N) t = (10,000kg) (24 m/s) t = 2s

13. I could push softly (low force), and it might take a while Or I could push really hard, and stop it almost immediately Ways to Get Same Impulse • If an object is moving, it has a certain momentum. • If I stop it (no matter how I stop it), it has zero momentum • So no matter how I stop it, the change in momentum is the same, so the impulse is the same

14. Impulse in action… Example : A 100 kg person falling out of a window Change in momentum: 1000 kg m/s Impulse = F t … has to equal 1000 kg m/s A) …Concrete: F(t) (stopped in a small amount of time) B) …a Trampoline: F(t) (stopped in a large amount of time)

15. Impulse in action… How else do we minimize force by increasing the time of impact? A. padded dashboards B. seatbelts and airbags C. helmets D. bungee cords E. sand-filled barrels on the freeway

16. Impulse is symmetric • Newton’s 3rd law: force of club on ball = force of ball on club Impulse on ball J = Ft Impulse on club J = Ft Same force, Same time, So impulse is the same

17. Unit 6: Momentum • Momentum • Impulse • Conservation of Momentum

18. Conservation of Momentum • What does it mean to “conserve” something? • Conservation –means that there is the same amount of something before and after some event occurs.

19. Law of Conservation of Momentum The total momentum of any system of objects is not changed (if the system is isolated). Add up all the vectors 5:30 pm Tuesday Total (or net) p = 30 kg m/s South 3:01 pm Friday Total p = 30 kg m/s South

20. Conservation of Momentum We usually think about it before and after an event– generally a collision Mathematically: Total momentum before = Total momentum after Total p before = Total p after

21. Conservation of Momentum Problems • We’ll generally deal with two objects traveling in one dimension • Total momentum is the sum of the momentums of the two objects • Be careful with directions! Pick a + and - direction

22. Conservation of Momentum Examples of conserved momentum: • Throwing a wrench in space • Gun recoil • All types of collisions

23. Before After Write an expression for each object’s momentum Make them equal! Find the totals Solving Problems A 2kg croquet ball (ball 1) going 8m/s to the right hits a 2kg croquet ball going 3m/s to the left. After the collision, ball 1 is going 1m/s to the right. How fast is ball 2 going after the collision? p1 = 2 * 8 p2 = 2 * -3 (NEG!) p1 = 16 kg m/s p2 = -6 kg m/s Total p = 10 kg m/s to the right p1 = 2 * 1 p2 = 2 * v p1 = 2 kg m/s p2 = 2 * v Total p = 2 + 2v 10 kg m/s = 2 + 2v v = 4

24. Conservation of Momentum If it helps you, use this approach: p = p' p1 + p2 = p1' + p2' m1v1 + m2v2 = m1v1 ' + m2v2 '

25. Conservation of Momentum Type 1 Example: A 60 kg ice skater stands motionless on the ice holding a 2 kg stone. She throws the stone at 15 m/s. How fast will the skater recoil? Before Total Mass = 62 kg v = 0 m/s p = 0 kg m/s Total p = 0 kg m/s to the right After m1 = 60kg m2 = 2 kg p1 = 60 * v p2 = 2 * 15 p1 = 60v p2 = 30 Total p’ = 60v + 30 kg m/s 0 = 60v + 30 V = -0.5 m/s

26. Notice the Symmetry • Object 2 gained 30 kg m/s • Object 1 lost 30 kg m/s • It will always work like this: • Δp1 = -Δ p2 • Whatever momentum one gains, the other loses After p1 = 60v p2 = 30 Total p’ = 60v + 30 kg m/s Before p = 0 kg m/s Total p = 0 kg m/s to the right 0 = 60v + 30 V = -0.5 m/s

27. Conservation of Momentum Type 2 Example: Two trucks collide and lock together as shown. Find the speed of the wreckage after colliding. 6 m/s 10 m/s 10,000kg 6,000 kg After Total mass = 16,000 kg p = 16,000 * v Total p’ = 16,000 v Before p1 = 6000 *10 p2 = 10000 * 6 p1 = 60,000 p2 = 60,000 Total p = 120,000 kg m/s 120,000 = 16,000 v V = 7.5 m/s

28. Things to be aware of for test questions on conservation • “Magnitude”: means you don’t need to worry about direction (+/-) • Most questions have zero momentum (“at rest”, “come to a stop”) • Some simple ones: just knowing that momentum is conserved (no numbers)

29. Conservation of Momentum Type 3 Example: After the break, the cue ball (1 kg) is rolling right at 3 m/s. It collides with the .7 kg 8-ball rolling left at 2 m/s. If the 8-ball bounces to the right at 2.5 m/s, what is the velocity and direction of the cue ball? Before p1 = 1 * 3 p2 = 0.7 * (-2) p1 = 3 p2 = -1.4 Total p = 1.6 kg m/s After p1 = 1 * v p2 = 0.7 * (-2) p1 = 1 * v p2 = -1.4 Total p = v – 1.4 kg m/s 1.6 = v -1.4 v = 3 kg m/s

30. Review • p = mv • p is in same direction as v • J = Ft = Δp so J = Ft = mΔv • Impulse is equal and opposite • Momentum is conserved • Total p before = total p after • Δp1 = -Δ p2