Chapter 9

1. Impulse & Momentum Chapter 9

2. Momentum • Momentum is what Newton called the “quantity of motion” of an object.

3. Momentum • The momentum of an object: • Depends on the object’s mass. • Momentum is directly proportional to mass. • Depends on the object’s velocity. • Momentum is directly proportional to velocity.

4. Momentum • In symbols: p = mv p v m

5. Momentum • Momentum is a vector quantity. • Common units of momentum: kg m/s

6. Impulse • The impulse exerted on an object depends on: • The force acting on the object. • Impulse is directly proportional to force. • The time that the force acts. • Impulse is directly proportional to time.

7. Impulse • In symbols: I = Ft I t F

8. Impulse • Impulse is a vector quantity. • Common units of impulse: N s

9. Impulse & Momentum • The impulse exerted on an object equals the object’s change in momentum.

10. Impulse & Momentum • In symbols: I = Dp

11. Conservation of Momentum • Since impulse = change in momentum, If no impulse is exerted on an object, the momentum of the object will not change.

12. Conservation of Momentum • If no external forces act on a system, the total momentum of the system will not change. • Such a system is called an “isolated system”.

13. Conservation of Momentum • Momentum is conserved in everyisolated system.

14. Conservation of Momentum • Another way to think about it is: Internal forces can never change the total momentum of a system.

15. Conservation of Momentum • In practice, for any event in an isolated system: • Momentumafter = Momentumbefore

16. What does a rocket push against? • Cars push on the road • Boats push on the water • Propellers push against air • Jet engines push air through turbines, heat it, and push against the hot exhaust (air) • What can you push against in space?

17. Momentum is conserved! • Before • After v = 0 so p = 0 M m v1 v2 m M pafter = Mv1 + mv2 = 0 as well so v1 = - (m/M) v2

18. A Rocket Engine: The Principle • Burn Fuel to get hot gas • hot = thermally fast  more momentum • Shoot the gas out the tail end • Exploit momentum conservation to accelerate rocket

19. A Rocket Engine: The Principle • Burn Fuel to get hot gas • Shoot the gas out the tail end • Exploit momentum conservation to accelerate rocket

20. Rockets push against the inertia of the ejected gas! • Imagine standing on a sled throwing bricks. • Conservation of momentum, baby! • Each brick carries away momentum, adding to your own momentum • Can eventually get going faster than you can throw bricks! • In this case, a stationary observer views your thrown bricks as also traveling forward a bit, but not as fast as you are

21. What counts? • The “figure of merit” for propellant is the momentum it carries off, mv. • It works best to get the propulsion moving as fast as possible before releasing it • Converting fuel to a hot gas gives the atoms speeds of around 6000 km/h! • Rockets often in stages: gets rid of “dead mass” • same momentum kick from propellant has greater impact on velocity of rocket if the rocket’s mass is reduced

22. Spray Paint Example • Imagine you were stranded outside the space shuttle and needed to get back, and had only a can of spray paint. Are you better off throwing the can, or spraying out the contents? Why? • Note: Spray paint particles (and especially the gas propellant particles) leave the nozzle at 100-300 m/s (several hundred miles per hour)

23. Going into orbit • Recall we approximated gravity as giving a const. acceleration at the Earth’s surface • It quickly reduces as we move away from the sphere of the earth • Imagine launching a succession of rockets upwards, at increasing speeds • The first few would fall back to Earth, but eventually one would escape the Earth’s gravitational pull and would break free • Escape velocity from the surface is 11.2 km/s

24. Going into orbit, cont. • Now launch sideways from a mountaintop • If you achieve a speed v such that v2/r = g, the projectile would orbit the Earth at the surface! • How fast is this? R ~ 6400 km = 6.4106 m, so you’d need a speed of sqrt[(6.4106m)(10m/s2)] = sqrt (6.4107) m/s, so: • v  8000 m/s = 8 km/s = 28,800 km/hr ~ 18,000 mph

25. 4 km/s: Not Fast Enough....

26. 6 km/s: Almost Fast Enough....but not quite!

27. 8 km/s: Not Too Fast, Nor Too Slow....Just Right

28. 10 km/s: Faster Than Needed to Achieve Orbit

29. Momentum

30. Angular momentum • Angular momentum is a product of a rotating objects moment of inertia and angular velocity • L = I  • kg x m2/s • I = mr2 • Moment of Inertia = mass times the distance from the axis squared

31. Angular momentum • Angular momentum is a product of a rotating objects moment of inertia and angular velocity • L = I  • kg x m2/s

32. Conservation of Angular momentum • If no net external torque acts on a closed system, then its angular momentum does not change • Li = Lf

33. Conservation of Angular momentum • If no net external torque acts on a closed system, then its angular momentum does not change • Li = Lf

34. Conservation of Angular momentum • If no net external torque acts on a closed system, then its angular momentum does not change • Li = Lf

35. Conservation of Angular momentum • If no net external torque acts on a closed system, then its angular momentum does not change • Li = Lf

36. Torque