Chapter 9

1. Chapter 9 Linear Momentum and Collisions

2. Intro • Consider bowling: • Bowling ball collides with initial pin • Force on/Acceleration of the Pin • Force on/Acceleration of the ball • Momentum- simplified way to study these moving objects.

3. 9.1 Linear Momentum and Conservation Consider two particles (isolated) m1 and m2 moving at v1 and v2 From Newton’s 3rd Law

4. 9.1 • And if the the masses are constant • If the derivative of a function is 0 it is constant (conserved)

5. 9.1 • Linear Momentum- the product of the mass and velocity of a moving particle. • Momentum is a vector quantity • Has dimensions MLT-1 and SI units kg.m/s • All momentum is conserved • Three Components of Momentum

6. 9.1 • Directly related to Newton’s 2nd Law • Instead of Net Force equals mass times accel. • Can be described as Net Force equals time rate of change of momentum

7. 9.1 • Conservation of Momentum • For isolated systems the time derivative of the total momentum is 0 • The total momentum is therefore constant or conserved. Law of Conservation of Momentum • And in 3 components

8. 9.1 • Quick Quizzes p 254-255 • Examples 9.1-9.2

9. 9.2 Impulse and Momentum • The momentum of an object changes when a net force acts on it or • Integrating this gives

10. 9.2 • Impulse-Momentum Theorem- the impulse of the force F acting on a particle equals the change in momentum of the particle • Force Varies, impulse time is short impulse can generally be calculated with the average force.

11. 9.2 • Quick Quizzes p. 258 • Examples 9.3-9.4

12. 9.3 Collisions in 1-D • Momentum is conserved • Three types of Collisions • Inelastic • Perfectly Inelastic • Elastic

13. 9.3 • Inelastic Collision- Momentum is conserved but kinetic energy is not. • Inelastic- objects collide and separate, some K is lost • Perfectly Inelastic- objects collide and stick together (moving as one), some K is lost

14. 9.3 • Elastic Collisions-A collision in which no energy is lost to (surroundings / internal / potential) • Both momentum and kinetic energy are conserved

15. 9.3 • By combining the Conservation of p and K equations • In all collision types careful attention to the direction (and sign) of velocities must be paid.

16. 9.3 • Quick Quizzes p. 262 • Examples 9.5 - 9.9

17. 9.4 2-D Collisions • Momentum is conserved on each axis • Examples 9.10 – 9.12

18. 9.5 Center of Mass • We can describe the overall motion of a mechanical system by tracking its center of mass • System could be a group of particles • System could be a large extended object • A force applied to the center of mass will cause no rotation to the system

19. 9.5 • To find the center of mass in 3-D space for a number (i) particles • Or in terms of the position vector of each particle

20. 9.5 • For extended objects that have a continuous mass distribution • Consider them an infinite number of closely spaced particles • The sum becomes an integral

21. 9.5 • Or in terms of the position vector • For symmetrical objects, the center of mass lies on the axis/plane of symmetry • Examples: uniform rod, sphere, cube, donut?

22. 9.5 • For extended objects, the force of gravity acts individually on each small piece of mass (dm) • The net effect of all these forces is equivalent to the single force Mg, through a point called the center of gravity. • If the gravitational field is uniform across all dm, the center of gravity and center of mass are one and the same.

23. 9.5 • Quick quiz p 272 • Examples 9.13, 9.14, 9.15

24. 9.6 Motion of a System • If the mass of a system remains constant (no particles entering/leaving) then we can track the motion of the center of mass, rather than the individual particles. • Also assumes any forces on the system are internal (isolated)

25. 9.6 • Velocity of the center of mass • Acceleration of the center of mass

26. 9.6 • If there is a net force on the system, it will move equivalent to the way a single M with the same net force would move. • And if the net force is zero

27. 9.6 • Quick Quizzes p. 276 • Examples 9.17, 9.18

28. 9.7 Rocket Propulsion • Most forms of vehicular motion result from action/reaction friction. • A rocket has nothing to push against so its motion/control depend on conservation of motion of the system. • The system includes the rocket body (and payload) plus the ejected fuel

29. 9.7 • The rocket burns fuel and oxidizer creating expanding gases that are directed through the nozzle. • Each gas molecule has a mass (that was once part of the rockets total mass) and velocity, therefore a downward momentum. • The rocket receives the same compensating momentum upward.

30. 9.7 • Looking a rocket initially with mass M + Δm, moving with velocity v…

31. 9.7 • And some time, Δt, later... • The rocket now has mass, M and velocity v + Δv, compensating the momentum of the exhausted mass, Δm.

32. 9.7 • The conservation of momentum expression for this change… • Can be simplified to…

33. 9.7 • A rocket motor produces a continuous flow of exhaust gas a fairly constant speed, through the burn • For continually changing values… Δv dv Δm dm So… 

34. 9.7 • Because the increase in exhaust mass = the decrease in rocket mass… • Then integrate this expression

35. 9.7 • Discuss integral of M-1 • Evaluating from vi to vf gives the basic expression for rocket propulsion.

36. 9.7 • Mi is the total mass of the rocket/payload plus fuel • Mf is the mass of the rocket/payload • Mi – Mf is the mass of fuel needed to achieve a certain speed (eg. Escape speed to power down rocket)

37. 9.7 • Thrust- the actual force on the rocket at any given time is • Thrust is proportional to exhaust speed and also the rate of change of mass (burn rate). • Examples 9.19 p. 279