1. Chapter 8 Conservation of Linear Momentum Linear momentum; Momentum conservation Impulse Total kinetic energy of a system March 9, 2010

2. Conservation of Linear Momentum Definition of linear momentum, Linear momentum is a vector (decompose to x,y,z directions). Units of linear momentum are kg-m/s. Can write Newton’s second law in terms of momentum: Momentum  force, as if Kinetic energy  work

3. Momentum of a system of particles The total momentum of a system of particles is the vector sum of the momenta of the individual particles: From Newton’s second law, we obtain

4. Conservation of Momentum Law of conservation of momentum: –If the sum of the external forces on a system is zero, the total momentum of the system does not change. If then Momentum is always conserved when no net “external” force. (even if “internal” forces are non-conservative).

6. Collisions “before” m1m1 m2m2 “after” m1m1 m2m2 momentum before collision = momentum after collision Always - But only if

7. Explosion - I “before” M “after” m1m1 m2m2 v1v1 v2v2 Example: m 1 = M/3 m 2 = 2M/3 After explosion, which block has larger momentum? (left, right, same)

8. “before” M “after” m1m1 m2m2 v1v1 v2v2 Example: m 1 = M/3 m 2 = 2M/3 After explosion, which block has larger momentum? (left, right, same) Each has the same momentum Explosion - I

9. “before” M “after” m1m1 m2m2 v1v1 v2v2 Example: m 1 = M/3 m 2 = 2M/3 After explosion, which block has larger momentum? (left, right, same) Each has the same momentum Which block has larger speed? Explosion - I

10. “before” M “after” m1m1 m2m2 v1v1 v2v2 Example: m 1 = M/3 m 2 = 2M/3 After explosion, which block has larger momentum? (left, right, same) Each has the same momentum Which block has larger speed? mv is the same for each block, so smaller mass has larger speed. Explosion - I

11. “before” M “after” m1m1 m2m2 v1v1 v2v2 Example: m 1 = M/3 m 2 = 2M/3 After explosion, which block has larger momentum? (left, right, same) Each has the same momentum Which block has larger speed? mv is the same for each block, so smaller mass has larger speed. Is kinetic energy conserved? Explosion - I

12. “before” M “after” m1m1 m2m2 v1v1 v2v2 Example: m 1 = M/3 m 2 = 2M/3 After explosion, which block has larger momentum? (left, right, same) Each has the same momentum Which block has larger speed? mv is the same for each block, so smaller mass has larger speed. Is kinetic energy conserved? NO! K was 0 before, it is greater after the explosion. (internal non-conservative force does some work.)

13. Momentum and Impulse l Momentum l For single object…. ç If F = 0, then momentum conserved (p = 0) For “system” of objects …

16. Elastic Collision in 1-Dimension Linear momentum is conserved Energy conserved (for elastic collision only) Initial Final

17. Elastic Collision Magnitude of relative velocity is conserved.

19. Is this an elastic collision? For elastic collision only:

20. Is this an elastic collision? Yes, the relative speeds are approximately the same before and after collision For elastic collision only:

21. What is the speed of the golf ball, in case of an elastic collision Club speed: 50 m/s Mass of clubhead: 0.5kg Mass of golfball: 0.05kg Two unknowns: after the impact, speed of club and speed of golfball Problem solving strategy: - Momentum conservation - Energy conservation (or use the derived equation for relative velocities)

22. Result: Special cases: 1) Golf shot: m1>>m2 Club speed almost unchanged Ball speed almost 2 x club speed 2) Neutron scatters on heavy nucleus: m1<<m2 neutron scatters back with almost same speed speed of nucleus almost unchanged

23. Some Terminology Elastic Collisions : collisions that conserve kinetic energy Inelastic Collisions: collisions that do not conserve kinetic energy * Completely Inelastic Collisons: objects stick together n.b. ALL CONSERVE MOMENTUM!! If external forces = 0

24. Kinetic energy of a system of particles: Where in terms of the CM velocity and relative velocity to the CM.