Welcome back to Physics 211

1. Welcome back to Physics 211 Today’s agenda: Conservation of momentum Elastic and inelastic collisions 2D problems

2. Reminder … • Professor Britton Plourde • Office = 223 Physics Building • Office hours = Tuesday, 2-4PM, or by appointment • Next Mastering Physics (MPHW 5) due Monday (11/7/05) by noon

3. Conservation of momentum(for a system consisting of two objects A and B) If the net (external) force on a system is zero, the total momentum of the system is constant. Whenever two or more objects in an isolated system interact, the total momentum of the system remains constant.

4. Points to note • Vector equation • Generalizes to arbitrary number of objects in system • True independent of nature of forces • True even when Newton’s laws fail!

5. Newton’s 3rd law and changes in momentum If all external forces (weight, normal, etc.) cancel:

6. If all external forces cancel, the changes in momentum of two interacting objects are equal and opposite:

7. Demo Two carts, initially at rest, tracked with motion detectors

8. A rocket is propelled by ejecting large amounts of (hot) gas at a very high speed (the exhaust speed, vex) relative to the rocket. Can the rocket ever obtain a speed that is greater than vex? 1. Yes 2. No 3. Not sure.

9. Rockets • Consider rocket with mass M at some instant ejects fuelDm with relative velocity vex. • New speed of rocket is v+Dv By Conservation of Momentum • If burn rate constant at b Dm = bDt 

10. Rocket equation continued • But M = MR + MF- bt  a = bvex/(MR+MF-bt) once all fuel is burnt …  • vF > vex if MF/MR+1 > e (2.718...)

11. A compact car (mC = 800 kg) and a large truck (mT = 12,000 kg) collide head-on on a slippery road. Which vehicle undergoes the larger change in momentum during the collision? 1. The car. 2. The truck. 3. The change in momentum is the same for both vehicles. 4. It is not possible to decide without knowing the initial speed of each vehicle.

12. Collisions If two objects collide and the net force exertedon the system (consisting of the two objects) iszero, the sum of their momenta is constant. The sum of their kinetic energies may or may not be constant.

13. Elastic and inelastic collisions • If K is conserved – collision is said to be elastic e.g. cue balls on a pool table • Otherwise termed inelastic e.g. lump of putty thrown against wall • Extreme case = completely inelastic -- objects stick together after collision

14. A cart moving to the right at speed v collides with an identical stationary cart on a low-friction track. The two carts stick together after the collision and move to the right. What is their speed after colliding? 1. 0.25 v 2. 0.5 v 3. v 4. 2v

15. Demo Completely inelastic collision between two carts tracked with motion detector

16. A cart (of mass m) moving to the right at speed v collides with an identical stationary cart on a low-friction track. The two carts stick together after the collision and move to the right with speed 0.5v In this collision, is mechanical energy conserved? 1. Yes 2. No 3. Depends on the actual values for m and v. 4. “Not sure.”

17. Cart A moving to the right at speed v collides with an identical stationary cart (cart B) on a low-friction track. This time the collision is elastic (i.e., there is no loss of kinetic energy of the system). What is each cart’s velocity after colliding (considering velocities to the right as positive)?

18. Check conservation of momentum and energy

19. Elastic collision of two masses v1i v1f v2f v2i = 0 m1 m2 m1 m2 Momentum m1v1i + 0 = m1v1f + m2v2f Energy  (1/2)m1v1i2 + 0= (1/2)m1v1f2 + (1/2)m2v2f2

20. Elastic collision of two masses v1i v2i = 0 v1f v2f m1 m2 m1 m2 m2v2f = m1(v1i - v1f) v2f = v1i + v1f

21. Special cases: (i) m1 = m2 v2f v2i = 0 v1i v1f m1 m2 m1 m2

22. Special cases: (ii) m1 << m2 v2f v1f v2i = 0 v1i m1

23. Demo Elastic collisions between two carts tracked with motion detectors

24. Relative velocities for elastic collisions • For example, velocity of m2 relative to m1 = v2 - v1 • With m2 stationary initially, • In general, for elastic collisions, relative velocity has same magnitude before and after collision

25. A student is sitting on a low-friction cart and is holding a medicine ball. The student then throws the ball at an angle of 60° (measured from the horizontal) with a speed of 10 m/s. The mass of the student (with the car) is 80 kg. The mass of the ball is 4 kg. What is the final speed of the student (with car)? 1. 0 m/s 2. 0.25 m/s 3. 0.5 m/s 4. 1 m/s

26. Momentum is a vector! • Must conserve components of momentum simultaneously • In 2 dimensions:

27. Example problem At the intersection of Texas Avenue and University Drive, a blue, subcompact car with mass 950 kg traveling east on University collides with a maroon pickup truck with mass 1900 kg that is traveling north on Texas and ran a red light. The two vehicles stick together as a result of the collision and, after the collision, the wreckage is sliding at 16.0 m/s in the direction 24o east of north. Calculate the speed of each vehicle before the collision. The collision occurs during a heavy rainstorm; you can ignore friction forces between the vehicles and the wet road. (Problem 8.34 from Young & Freedman)

28. Example problem Two pucks undergo an elastic collision on a frictionless table. Puck A has mass mA = 0.500 kg and puck B has mass mB = 0.300 kg. Puck A has an initial velocity of 4.00 m/s in the positive x-direction and a final velocity of 2.00 m/s in an unknown direction. Puck B is initially at rest. Find the final speed vB2 of puck B and the angles  and . (Example 8.13 from Young & Freedman)

29. Demo Newton’s cradle with bowling balls

30. In the demonstration called Newton’s Cradle balls 1 and 2 are pulled back and released. They hit the other balls with speed v. Which of the following outcomes are consistent with conservation of momentum and conservation of mechanical energy?

31. Check conservation of momentum and energy