Lecture 12 Momentum, Energy, and Collisions

1. Lecture 12 Momentum, Energy, and Collisions

2. Announcements • EXAM: Thursday, October 17 (chapter 6-9) • Equation sheet to be posted • Practice exam to be posted

3. Impulse Linear Momentum With no net external force:

4. Center of Mass The center of mass of a system is the point where the system can be balanced in a uniform gravitational field. For two objects: The center of mass is closer to the more massive object. Think of it as the “average location of the mass”

5. (1) (2) XCM Center of Mass a) higher b) lower c) at the same place d) there is no definable CM in this case The disk shown below in (1) clearly has its center of mass at the center. Suppose the disk is cut in half and the pieces arranged as shown in (2).Where is the center of mass of(2) ascompared to(1) ?

6. (1) (2) CM XCM Center of Mass a) higher b) lower c) at the same place d) there is no definable CM in this case The disk shown below in (1) clearly has its center of mass at the center. Suppose the disk is cut in half and the pieces arranged as shown in (2). Where is the center of mass of(2) ascompared to(1) ? The CM of each half is closer to the top of the semicirclethan the bottom. The CM of the whole system is located at themidpoint of the two semicircle CMs, which ishigherthan the yellow line.

7. Gun Control When a bullet is fired from a gun, the bullet and the gun have equal and opposite momenta. If this is true, then why is the bullet deadly (whereas it is safe to hold the gun while it is fired)? a) it is much sharper than the gun b) it is smaller and can penetrate your body c) it has more kinetic energy than the gun d) it goes a longer distance and gains speed e) it has more momentum than the gun

8. Gun Control When a bullet is fired from a gun, the bullet and the gun have equal and opposite momenta. If this is true, then why is the bullet deadly (whereas it is safe to hold the gun while it is fired)? a) it is much sharper than the gun b) it is smaller and can penetrate your body c) it has more kinetic energy than the gun d) it goes a longer distance and gains speed e) it has more momentum than the gun Even though it is true that the magnitudes of the momenta of the gun and the bullet are equal, the bullet is less massive and so it has a much higher velocity. Because KE is related to v2, the bullet has considerably more KE and therefore can do more damage on impact.

9. Rolling in the Rain a) speeds up b) maintains constant speed c) slows down d) stops immediately An open cart rolls along a frictionless track while it is raining. As it rolls, what happens to the speed of the cart as the rain collects in it? (Assume that the rain falls vertically into the box.)

10. Rolling in the Rain a) speeds up b) maintains constant speed c) slows down d) stops immediately An open cart rolls along a frictionless track while it is raining. As it rolls, what happens to the speed of the cart as the rain collects in it? (Assume that the rain falls vertically into the box.) Because the rain falls in vertically, it adds no momentum to the box, thus the box’s momentum is conserved. However, because the mass of the box slowly increases with the added rain, its velocity has to decrease.

11. initial final px = mv0 px = (2m)vf mass m mass m Two objects collide... and stick No external forces... so momentum of system is conserved mv0 = (2m)vf vf = v0 / 2 A completely inelastic collision: no “bounce back”

12. Kinetic energy is lost! KEfinal = 1/2 KEinitial mass m mass m Inelastic collision: What about energy? vf = v0 / 2 initial final

13. Collisions This is an example of an “inelastic collision” Collision: two objects striking one another Elastic collision⇔ “things bounce back” ⇔ energy is conserved Inelastic collision: less than perfectly bouncy ⇔ Kinetic energy is lost Time of collision is short enough that external forces may be ignored so momentum is conserved Completely inelastic collision: objects stick together afterwards. Nothing “bounces back”. Maximal energy loss

14. Completely inelastic collision: colliding objects stick together, maximal loss of kinetic energy Elastic collision: momentum and kinetic energy is conserved. Elastic vs. Inelastic Inelastic collision: momentum is conserved but kinetic energy is not

15. Momentum Conservation: Completely Inelastic Collisions in One Dimension Solving for the final momentum in terms of initial velocities and masses, for a 1-dimensional, completely inelastic collision between unequal masses: Completely inelastic only (objects stick together, so have same final velocity) KEfinal < KEinitial

16. Crash Cars I a) I b) II c) I and II d) II and III e) all three If all three collisions below are totally inelastic, which one(s) will bring the car on the left to a complete halt?

17. Crash Cars I a) I b) II c) I and II d) II and III e) all three If all three collisions below are totally inelastic, which one(s) will bring the car on the left to a complete halt? In case I, the solid wall clearly stops the car. In cases II and III, becauseptot = 0 before the collision, thenptot must also be zero after the collision, which means that the car comes to a halt in all three cases.

18. Crash Cars II a) I b) II c) III d) II and III e) all three If all three collisions below are totally inelastic, which one(s) will cause the most damage (in terms of lost energy)?

19. The car on the left loses the same KE in all three cases, but incase III, the car on the right loses the most KE becauseKE = mv2 and the car incase IIIhas thelargest velocity. Crash Cars II a) I b) II c) III d) II and III e) all three If all three collisions below are totally inelastic, which one(s) will cause the most damage (in terms of lost energy)?

20. momentum conservation in inelastic collision PE = (m+M) g h energy conservation afterwards hmax = (mv0)2 / [2 g (m+M)2] KE = 1/2 (mv0)2 / (m+M) vf = m v0 / (m+M) Ballistic pendulum: the height h can be found using conservation of mechanical energy after the object is embedded in the block.

21. approximation Velocity of the ballistic pendulum Pellet Mass (m): 2 g Pendulum Mass (M): 3.81 kg Wire length (L): 4.00 m

22. Momentum is a vector equation: there is 1 conservation of momentum equation per dimension Energy is not a vector equation: there is only 1 conservation of energy equation Inelastic Collisions in 2 Dimensions For collisions in two dimensions, conservation of momentum is applied separately along each axis:

23. Elastic Collisions In elastic collisions, both kinetic energyandmomentum are conserved. One-dimensional elastic collision:

24. Note: relative speed is conserved for head-on (1-D) elastic collision We have two equations: solving for the final speeds: conservation of momentum conservation of energy and two unknowns (the final speeds). Elastic Collisions in 1-dimension For special case of v2i = 0

25. Limiting cases of elastic collisions note: relative speed conserved

26. Limiting cases note: relative speed conserved

27. Limiting cases note: relative speed conserved

28. Incompatible! Toy Pendulum Could two balls recoil and conserve both momentum and energy?

29. at rest v at rest v 1 2 Elastic Collisions I a) situation 1 b) situation 2 c) both the same Consider two elastic collisions: 1) a golf ball with speed v hits a stationary bowling ball head-on. 2) a bowling ball with speed v hits a stationary golf ball head-on. In which case does the golf ball have the greater speed after the collision?

30. v 2v v 1 2 Elastic Collisions I a) situation 1 b) situation 2 c) both the same Consider two elastic collisions: 1) a golf ball with speed v hits a stationary bowling ball head-on. 2) a bowling ball with speed v hits a stationary golf ball head-on. In which case does the golf ball have the greater speed after the collision? Remember that the magnitude of the relative velocity has to be equal before and after the collision! In case1the bowling ball will almost remain at rest, and thegolf ball will bounce back with speed close to v. In case2the bowling ball will keep going with speed close to v, hence thegolf ballwill rebound with speed close to 2v.

31. A charging bull elephant with a mass of 5240 kg comes directly toward you with a speed of 4.55 m/s. You toss a 0.150 kg rubber ball at the elephant with a speed of 7.81 m/s. When the ball bounces back toward you, what is its speed?

32. In the frame, what is the final speed of the ball? Back in the frame of the ground: Our simplest formulas for speed after an elastic collision relied on one body being initially at rest. So lets try a frame where one body (the elephant) is at rest! What is the speed of the ball relative to the elephant? A charging bull elephant with a mass of 5240 kg comes directly toward you with a speed of 4.55 m/s. You toss a 0.150 kg rubber ball at the elephant with a speed of 7.81 m/s. When the ball bounces back toward you, what is its speed?

33. In the frame, what is the final speed of the ball? Back in the frame of the ground: Our simplest formulas for speed after an elastic collision relied on one body being initially at rest. So lets try a frame where one body (the elephant) is at rest! What is the speed of the ball relative to the elephant? A charging bull elephant with a mass of 5240 kg comes directly toward you with a speed of 4.55 m/s. You toss a 0.150 kg rubber ball at the elephant with a speed of 7.81 m/s. When the ball bounces back toward you, what is its speed? NOTE: Formulas for 1-D elastic scattering with non-zero initial velocities are given in end-of-chapter problem 88.

34. Elastic Collisions in 2-D Two-dimensional collisions can only be solved if some of the final information is known, such as the final velocity of one object

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37. v1 v0 = 3.5x105 m/s 37o 53o initial v2 final A proton collides elastically with another proton that is initially at rest. The incoming proton has an initial speed of 3.5x105 m/s and makes a glancing collision with the second proton. After the collision one proton moves at an angle of 37o to the original direction of motion, the other recoils at 53o to that same axis. Find the final speeds of the two protons.

38. v1 v0 = 3.5x105 m/s 37o 53o initial v2 final A proton collides elastically with another proton that is initially at rest. The incoming proton has an initial speed of 3.5x105 m/s and makes a glancing collision with the second proton. After the collision one proton moves at an angle of 37o to the original direction of motion, the other recoils at 53o to that same axis. Find the final speeds of the two protons. Momentum conservation: if we’d been given only 1 angle, would have needed conservation of energy also!

39. Rotational Kinematics

40. Angular Position

41. θ > 0 θ < 0 Angular Position Degrees and revolutions:

42. 1 complete revolution = 2 π radians C = 2 π r C / D = π 1 rad = 360o / (2π) = 57.3o Arc Length Arc length s, from angle measured in radians: s = rθ - What is the relationship between the circumference of a circle and its diameter? - Arc length for a full rotation (360o) of a radius=1m circle? s = 2 π (1 m) = 2 π meters

43. Angular Velocity

44. Period = How long it takes to go 1 full revolution Period T: SI unit: second, s Instantaneous Angular Velocity

45. Linear and Angular Velocity

46. Greater translation for same rotation

47. Bonnie and Klyde II a) Klyde b) Bonnie c) both the same d) linear velocity is zero for both of them Bonnie sits on the outer rim of amerry-go-round, and Klyde sits midway between the center and the rim. The merry-go-round makes one revolution every 2 seconds. Who has the larger linear (tangential) velocity? ω Bonnie Klyde

48. ω Bonnie and Klyde II Bonnie sits on the outer rim of a merry-go-round, and Klyde sits midway between the center and the rim. The merry-go-round makes one revolution every 2 seconds. Who has the larger linear (tangential) velocity? a) Klyde b) Bonnie c) both the same d) linear velocity is zero for both of them Their linear speeds v will be different because v = r ω and Bonnie is located farther out (larger radius r) than Klyde. Klyde Bonnie

49. Angular Acceleration

50. Instantaneous Angular Acceleration