Unit 5: Conservation of Momentum

1. Unit 5: Conservation of Momentum • Force and Momentum and Conservation of Momentum (9-1,9-2, 9-3) • Collisions (9-4, 9-5, 9-6) • Collisions, Center of Mass (9-7,9-8) • Catch-up, Rocket Propulsion, and Quiz Three Review (9-9,9-10) Physics 253

2. Schedule • No class Friday, March 30th! • Next test Friday, April 6th • Please, work on your problems sets and extra credit. • If you need information to compute your grade send email or see me at my office! Physics 253

3. Conservation Laws • There are many conservation laws, we’ve already discussed Conservation of Energy. • In this course we will also discuss • Conservation of Momentum • Conservation of Angular Momentum • Many others: charge, baryon number, lepton number…all are a consequence of some fundamental symmetry of nature. • When we combine the conservation of energy, momentum, and angular momentum we can beautifully describe complex systems of objects. Physics 253

4. Momentum • Linear momentum is a deceptively simple quantity equal to the product of an object’s mass, m, and velocity, v: p=mv • Properties of momentum • A vector with same direction as the velocity, which requires a reference frame • Magnitude equal to mv, increases linearly with m and with velocity. • SI unit is kg-m/s but carries no explicit name. • Newton actually did have a name he called it “the quantity of motion”. (Easy to see why it didn’t stick!) Physics 253

5. A force is required to change magnitude or direction of momentum with respect to time. Actually this is similar to Newton’s original statement of the 2nd Law: The rate of change of momentum of an object is equal to the net force applied to it. Or in symbols: A quick derivation shows the two versions of the 2nd law are equivalent: Note we assumed that the mass was constant! Actually the equality of the force to the change of momentum is more general and useful. For instance, if we allow the mass to change we can describe propulsion … Force and Momentum Physics 253

6. Example: Water Hitting a Car • Water leaves a hose and hits a car at a rate of 1.5 kg/s and a speed of +20 m/s. • What is the force exerted by the water on the car? • What if the water splashes off at -5m/s? Is the force greater or less? Physics 253

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9. Experimental Conservation of Momentum • Consider head-on collision of two hard balls • Assume no net external forces. • In the 17th century and predating Newton it was found that: • Individual momentum can change • Vector sum of momentum was observed to be constant. Physics 253

10. Theoretical Conservation of Momentum • The experimental result can be explained using Newton’s Laws. • Consider two colliding objects • Initial momentum p1 and p2 • Final momentum p’1 and p’2 • Object 1 exerts a force F on object 2 • Object 2 exerts force –F on object 1. • No other significant forces involved. Physics 253

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12. Example: Goal Line Stand • A 90-kg fullback attempts to dive over the goal line with a velocity of 6.00 m/s. He is met at the goal line by a 110-kg linebacker moving at 4.00 m/s in the opposite direction.  The linebacker holds on to the fullback.  • Does the fullback cross the goal line? Physics 253

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14. Example: The Kick of a Rifle • What is the recoil velocity of a 5.0-kg rifle that shoots at 0.050-kg bullet at a muzzle velocity of 120m/s? • What would be the initial velocity of a 75-kg rifleperson? Physics 253

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16. Example: Billiard Ball Collision in 2-D • A billiard ball moving at 3.0m/s in the x direction strikes a ball of equal mass initially at rest. The two balls move off at 45o wrt to the x axis as shown. • What are the speeds of the two balls after the collision? Physics 253

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19. Generalization to a System of Objects • The derivation can be extended to any number of objects. • The total initial momentum is: P = m1v1+m2v2+…+mnvn=Smivi = Spi • Which can be differentiated with respect to time: dP/dt = d(Spi)/dt = SFi • Where Fi is the net force on the ith object. There are internal & external forces. But remember that internal forces come in equal and opposite pairs so they will cancel in the summation. Thus: dP/dt = SFexternal • Now if the net external forces are zero then the change in momentum of the system is zero and P=constant! Physics 253

20. The Law of Conservation of Momentum: The total momentum of an isolated system of bodies remains constant. Physics 253

21. Example: An Exploded Firecracker • A firecracker with a mass of 100g, initially at rest, explodes into 3 parts.   One part with a mass of 25g moves along the x-axis at 75m/s.  One part with mass of 34g moves along the y-axis at 52m/s.  • What is the velocity of the third part?  Physics 253

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23. Impulse • Billiard balls interact almost instantaneously, certainly in a fraction of a second. • As shown in the figures, when two objects interact the contact force rises rapidly from zero to a maximum and just as quickly falls to zero. • This occurs in a small time interval Dt, during which there is an impulse of force. Physics 253

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25. Properties of Impulse • Units are N-s or kg-m/s. • Equal to the area under the force versus time curve. • Convenient to calculate in terms of the average force during an event. Where the average force is defined as the constant force which, if acting over the time interval of the interaction would produce the same impulse and momentum change: Physics 253

26. Example: The Impulse of a Jump • Calculate the impulse on a 70-kg person when landing on the ground after jumping from 3.0m. • Estimate the average force exerted on the person’s feet by the ground if landing is • Stiff legged (body moves 0.01m) • With bent knees. (body moves (0.50m) Physics 253

27. Remember: We can’t get the impulse by the time integral of the force, but we can get it by calculating the momentum. The final velocity is zero! The initial velocity can be determined by using energy conservation: Physics 253

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29. Next • Collisions: Elastic, Inelastic, 2 and 3Dimensional • No class Friday, March 30th! • Next test Friday, April 6th Physics 253