Conservation of Linear Momentum in Two-Dimensional Collisions

1. Chapter 8Conservation of Linear MomentumTwo dimensional collisionsVarying mass and rocket propulsion Second midterm exam Next Thursday, March 18 at 5:45 -7PM. Chapts 5-8.If you have a conflict, email to BOTH than@hep.wisc.edu; onellion@wisc.edu To sign up the alternative at 3:30-4:45pm, Thur, March 18 Same exam rooms as the last  Please consult with the course web.Practice exams are available on the Web. TA’s will hold review sessions. March 11, 2010

2. Ballistic Pendulum What is the initial velocity vli of the projectile? Known quantities: m1, m2, h • Two stage process: 1. m1collides with m2, completely inelastically. Both m2 and m1 then move together with a velocity Vf (before having risen significantly). 2. Both (m1 + m2) rise a height h, conserving energy E. (no non-conservative forces acting after collision)

3. Ballistic Pendulum • Stage 1: Momentum is conserved Energy is not conserved in x-direction: • Stage 2: Energy is conserved Substituting for Vgives:

4. y x before after Collisions or Explosions in Two Dimensions • Ptotal,xandPtotal,yindependently conserved • Ptotal,x,before = Ptotal,x,after • Ptotal,x,before = Ptotal,x,after

5. B A M Explosions “before” “after” Which of these is possible? A (p appears conserved) B (p not conserved in y direction) both

6. B A M Explosions “before” “after” Which of these is possible? A (p not conserved in y direction) B neither

7. Shooting Pool... • Assuming • Collision is elastic (KE is conserved) • No spin is imparted • Balls have the same mass • One ball starts out at rest • Then the angle between the balls after the collision is 90o pf pi vcm Pf F before after

8. p1 p2 p0 y p1 q x f p2 Shooting Pool • Elastic means conservation of kinetic energy This is the Pythagorean formula for arighttriangle • Momentum conservation: Choose axes such that the initial momentum is in x direction, then • Then we should have: • + f = 900 Then cos q = sin fand cos f = sin q Px = p0 Initial: Py = 0

9. Shooting Pool... • Tip: If you shoot a ball spotted on the “dot”, you have a good chance of scratching !

10. Suppose rain falls vertically into an open cart rolling with negligible friction along a straight horizontal track. As a result of the accumulating water, the speed of the cart • increases.   • does not change. • decreases. • Mass is increasing - vertical impulse by track - inelastic collision • Mass is increasing and the horizontal momentum is fixed • So the velocity must decrease! • and so does kinetic energy(p2/2M)

11. Continuously variable mass • At t: M with v; plus ΔM with u • At t+Δt: M+ΔM with v+Δv. Thus the impulse

12. Continuously variable mass In terms of the “relative velocity” We arrive:

14. Rocket (jet) Propulsion • The operation of a rocket (jet) depends on the law of conservation of momentum as applied to a system, where the system is the rocket plus its ejected fuel • This is different than propulsion on the earth where two objects exert forces on each other • road on car • train on track • The rocket is accelerated as a result of the thrust of the exhaust gases • This represents the inverse of an inelastic collision • Momentum is conserved • Kinetic Energy is increased (like the explosion process, at the expense of the stored energy of the rocket fuel)

15. Rocket Propulsion The second law from last page: Define the “thrust”: Vertical direction: 