1. Chapter 8 Conservation of Linear Momentum Two dimensional collisions Varying mass and rocket propulsion
March 11, 2010
Second midterm exam Next Thursday, March 18 at 5:45 -7PM. Chapts If you have a conflict, to BOTH
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.
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2. Ballistic Pendulum What is the initial velocity vli of the projectile?
Known quantities:
m1, m2, h
Two stage process:
1. m1 collides 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)
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3. Ballistic Pendulum Stage 1: Momentum is conserved
Energy is not conserved
in x-direction:
Stage 2: Energy is conserved
Substituting for V gives:
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4. Collisions or Explosions in Two Dimensions
after y x
before
Ptotal,x and Ptotal,y independently conserved
Ptotal,x,before = Ptotal,x,after
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5. Explosions Which of these is possible? A (p appears conserved)M
“before” B
“after” A
Which of these is possible?
A (p appears conserved)
B (p not conserved in y direction)
both
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6. Explosions Which of these is possible?M
“before” A B
“after”
Which of these is possible?
A (p not conserved in y direction) B
neither
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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 F Pf
before
after
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8. Then cos q = sin f and cos f = sin q
Shooting Pool
Elastic means conservation of kinetic energy
This is the Pythagorean formula for a right triangle
Momentum conservation: p0 p2 p1 y x q f p1 p2
Choose axes such that the initial momentum is in x direction, then
Then we should have:
+ f = 900
Then cos q = sin f and cos f = sin q
Px = p0
Initial:
Py = 0
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9. Shooting Pool...
Tip: If you shoot a ball spotted on the “dot”, you have a good chance of scratching !
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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)
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11. Continuously variable mass
At t: M with v;
plus ΔM with u
At t+Δt: M+ΔM
with v+Δv.
Thus the impulse
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12. Continuously variable mass
In terms of the “relative velocity”
We arrive:
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13. 11/21/2018

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