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1. Linear Momentum Physics Montwood High School R. Casao

2. Linear Momentum • Linear momentum of an object is the mass of the object multiplied by its velocity. • Momentum: p = m·v • Unit: kg·m/s or N·s • Both momentum and kinetic energy describe the motion of an object and any change in mass and/or velocity will change both the momentum and kinetic energy of the object.

3. Linear Momentum • Momentum refers to inertia in motion. • Momentum is a measure of how difficult it is to stop an object; a measure of “how much motion” an object has. • More force is needed to stop a baseball thrown at 95 mph than to stop a baseball thrown at 45 mph, even though they both have thesame mass.

4. Linear Momentum • More force is needed to stop a train moving at 45 mph than to stop a car moving at 45 mph, even though they both have thesame speed. • Both mass and velocity are important factors when considering the force needed to change the motion of an object.

5. Impulse • Impulse (J) = force·time • Equation: J = F·t Unit: N·s • The impulse of a force is equal to the change in momentum of the body to which the force is applied. This usually means a change in velocity. F·t = m·v where v = vf - vi • The same change in momentum can be accomplished by a small force acting for a long time or by a large force acting for a short time.

6. Impulse • The area under the curve in a force vs. time graph represents the change in momentum (m·v). • If your car runs into a brick wall and you come to rest along with the car, there is a significant change in momentum. If you are wearing a seat belt or if the car has an air bag, your change in momentum occurs over a relatively long time interval. If you stop because you hit the dashboard, your change in momentum occurs over a very short time interval.

7. Impulse • If a seat belt or air bag brings you to a stop over a time interval that is five times as long as required to stop when you strike the dashboard, then the forces involved are reduced to one-fifth of the dashboard values. That is the purpose of seat belts, air bags, and padded dashboards. By extending the time during which you come to rest, these safety devices help reduce the forces exerted on you. • If you want to increase the momentum of an object as much as possible, you apply the greatest force you can for as long a time as possible.

8. A 1000 kg car moving at 30 m/s (p = 30,000 kg m/s) can be stopped by 30,000 N of force acting for 1.0 s (a crash!) • or by 3000 N of force acting for 10.0 s (normal stop)

9. Impulse and Bouncing • Impulses are greater when bouncing takes place. • The impulse required to bring an object to a stop and then throw it back again is greater than the impulse required to bring an object to a stop.

10. Conservation of Linear Momentum • In a closed system of objects, linear momentum is conserved as the objects interact or collide. The total vector momentum of the system remains constant. p before interaction = p after interaction

11. Perfectly Inelastic Collisions • Perfectly inelastic collisions are those in which the colliding objects stick together and move with the same velocity. • Kinetic energy is lost to other forms of energy in an inelastic collision.

12. Inelastic Collision Example • Cart 1 and cart 2 collide and stick together • Momentum equation: • Kinetic energy equation: v1 and v2 = velocities before collision v  = velocity after collision

13. Directions for Velocity • Momentum is a vector, so direction is important. • Velocities are positive or negative to indicate direction. • Example: bounce a ball off a wall

14. Inelastic Collisions • Kinetic energy is lost when the objects are deformed during the collision. • Momentum is conserved.

15. Elastic Collisions • Momentum and kinetic energy are conserved in an elastic collision. • The colliding objects rebound from each other with NO loss of kinetic energy.

16. Elastic Collision Example • Example: mass 1 and mass 2 collide and bounce off of each other • Momentum equation: • Kinetic energy equation: v1 and v2 = velocities before collision v1 and v2 = velocities after collision • Velocities are + or – to indicate directions.

17. Elastic Collisions Involving an Angle • Momentum is conserved in both the x-direction and in the y-direction. • Before:

18. Elastic Collisions Involving an Angle • Directions for the velocities before and after the collision must include the positive or negative sign. • The direction of the x-components for v1 and v2 do not change and therefore remain positive. • The directions of the y-components for v1 and v2 do change and therefore one velocity is positive and the other velocity is negative.

19. Elastic Collisions Involving an Angle • px before = px after • py before = py after • Velocity after collision:

20. Elastic Collisions • Perfectly elastic collisions do not have to be head-on. • Particles can divide or break apart. • Example: nuclear decay (nucleus of an element emits an alpha particle and becomes a different element with less mass)

21. Elastic Collisions • mn = mass of nucleus • mp = mass of alpha particle • vn = velocity of nucleus before event • vn’ = velocity of nucleus after event • vp = velocity of particle after event

22. Recoil • Recoil is the term that describes the backward movement of an object that has propelled another object forward. In the nuclear decay example, the vn’ would be the recoil velocity.

23. Head-on and Glancing Collisions • Head-on collisions occur when all of the motion, before and after the collision, is along one straight line. • Glancing collisions involve an angle. • A vector diagram can be used to represent the momentum for a glancing collision.

24. Vector Diagrams • Use the three vectors and construct a triangle.

25. Vector Diagrams • Use the appropriate expression to determine the unknown variable.

26. Vector Diagrams • Total vector momentum is conserved. You could break each momentum vector into an x and y component. px before = px after py before = py after • You would use the x and y components to determine the resultant momentum for the object in question • Resultant momentum =

27. Vector Diagrams • Right triangle trigonometry can be used to solve this type of problem:

28. Vector Diagrams • Pythagorean theorem: • If the angle  for the direction in which the cars go in after the collision is known, you can use sin, cos, or tan to determine the unknown quantity. Example: determine final velocity vT if the angle is 25°.

29. Vector Diagrams • To determine the angle at which the cars go off together after the impact:

30. Special Condition • When a moving ball strikes a stationary ball of equal mass in a glancing collision, the two balls move away from each other at right angles. • ma = mb • va = 0 m/s

31. Special Condition • Use the three vectors to construct a triangle.

32. Special Condition • Use the appropriate expression to determine the unknown variable.

33. Rocket Propulsion (Jet Propulsion) • As fuel burns and exhaust gases leave the rocket or jet engine, momentum is carried with them. • To conserve momentum, the rocket or jet must gain the same amount of momentum in the opposite direction.

34. Rocket Propulsion (Jet Propulsion) • Thrust = the magnitude (size) of the force exerted by an engine or rocket. • Mass flow rate for fuel = • The direction of the thrust is opposite to the direction of the exhaust gases coming from the rocket or jet engine.

35. Ballistic Pendulum • In the ballistic pendulum lab, a ball of known mass is shot into a pendulum arm. The arm swings upward and stops when its kinetic energy is exhausted. • From the measurement of the height of the swing, one can determine the initial speed of the ball. • This is an inelastic collision. As always, linear momentum is conserved.

36. Ballistic Pendulum

37. Ballistic Pendulum • Potential energy of ball in gun: • Ball embeds in pendulum:

38. Ballistic Pendulum • Pendulum rises to a maximum height: • Solving for the initial speed of the projectile we get:

39. Helpful Websites • Physics Classroom – Momentum and Collisions • ExploreScience – Two Dimensional Collisions

40. Elastic Collision Example • Example: mass 1 and mass 2 collide and bounce off of each other • Momentum equation: • Kinetic energy equation: v1 and v2 = velocities before collision v1 and v2 = velocities after collision • Velocities are + or – to indicate directions.

41. Elastic Collision Example • Working with kinetic energy: • 0.5 cancels out.

42. Elastic Collision Example • The velocity terms are perfect squares and can be factored: a2-b2 = (a – b)·(a + b) • We will use this equation later.

43. Elastic Collision Example • Momentum equation:

44. Elastic Collision Example • Both the kinetic energy and momentum equations have been solved for the ratio of m1/m2. • Set m1/m2 for kinetic energy equal to m1/m2 for momentum:

45. Elastic Collision Example • Get all the v1 terms together and all the v2 terms together: • Cancel the like terms:

46. Elastic Collision Example • Rearrange to get the initial and final velocities back together on the same side of the equation: • This equation can be solved for one of the two unknowns, then substituted back into the conservation of momentum equation.