Momentum and Momentum Conservation

1. Momentum and Momentum Conservation • Momentum • Impulse • Conservation of Momentum • Collision in 1-D • Collision in 2-D

2. So What’s Momentum ? • Momentum = mass x velocity • This can be abbreviated to : . momentum = mv • Or, if direction is not an important factor : . . momentum = mass x speed • So, A really slow moving truck and an extremely fast roller skate can have the same momentum.

3. 1 kg 10 m/sec 1000 kg .01 m/sec Question : • Under what circumstances would the roller skate and the truck have the same momentum ? • The roller skate and truck can have the same momentum if the ratio of the speed of the skate to the speed of the truck is the same as the ratio of the mass of the truck to the mass of the skate. • A 1000 kg truck moving at 0.01 m/sec has the same momentum as a 1 kg skate moving at 10 m/sec. Both have a momentum of 10 kg m/sec. ( 1000 x .01 = 1 x 10 = 10 )

4. Linear Momentum • A new fundamental quantity, like force, energy • The linear momentump of an object of mass m moving with a velocity is defined to be the product of the mass and velocity: • The terms momentum and linear momentum will be used interchangeably in the text • Momentum depend on an object’s mass and velocity

5. Newton’s Law and Momentum • Newton’s Second Law can be used to relate the momentum of an object to the resultant force acting on it • The change in an object’s momentum divided by the elapsed time equals the constant net force acting on the object

6. Impulse and Momentum • If momentum changes, it’s because mass or velocity change. • Most often mass doesn’t change so velocity changes and that is acceleration. • And mass x acceleration = force • Applying a force over a time interval to an object changes the momentum • Force x time interval = Impulse • Impulse = F t or Ft = mv Ft = mv

7. Impulse-Momentum Theorem • The theorem states that the impulse acting on a system is equal to the change in momentum of the system

8. MOMENTUM • An object at rest has no momentum, why? • Because anything times zero is zero • (the velocity component is zero for an object at rest) • To INCREASE MOMENTUM, apply the greatest force possible for as long as possible. • Examples : • pulling a sling shot • drawing an arrow in a bow all the way back • a long cannon for maximum range • hitting a golf ball or a baseball . (follow through is important for these !) FORCE TIME

9. MOMENTUM • SOME VOCABULARY : • impulse : impact force X time (newton.sec) . Ft = impulse • impact : the force acting on an object (N) . usually when it hits something. • impact forces : average force of impact

10. MOMENTUM • Decreasing Momentum • Which would it be more safe to hit in a car ? • Knowing the physics helps us understand why hitting a soft object is better than hitting a hard one. Ft mv mv Ft

11. MOMENTUM • In each case, the momentum is decreased by the same amount or impulse (force x time) • Hitting the haystack extends the impact time (the time in which the momentum is brought to zero). • The longer impact time reduces the force of impact and decreases the deceleration. • Whenever it is desired to decrease the force of impact, extend the time of impact !

12. DECREASING MOMENTUM • If the time of impact is increased by 100 times (say from .01 sec to 1 sec), then the force of impact is reduced by 100 times (say to something survivable). • EXAMPLES : • Padded dashboards on cars • Airbags in cars or safety nets in circuses • Moving your hand backward as you catch a fast-moving ball with your bare hand or a boxer moving with a punch. • Flexing your knees when jumping from a higher place to the ground. or elastic cords for bungee jumping • Using wrestling mats instead of hardwood floors. • Dropping a glass dish onto a carpet instead of a sidewalk.

13. EXAMPLES OF DECREASING MOMENTUM Ft = change in momentum • Bruiser Bruno on boxing … • Increased impact time reduces force of impact • Barbie on bungee Jumping … Ft = change in momentum POOF ! CRUNCH ! Ft = Δmv applies here. mv = the momentum gained before the cord begins to stretch that we wish to change. Ft = the impulse the cord supplies to reduce the momentum to zero. Because the rubber cord stretches for a long time the average force on the jumper is small.

14. Questions : • When a dish falls, will the impulse be less if it lands on a carpet than if it lands on a hard ceramic tile floor ? • The impulse would be the same for either surface because there is the same momentum change for each. It is the forcethat is less for the impulse on the carpet because of the greater time of momentum change. There is a difference between impulse and impact. • If a boxer is able to increase the impact time by 5 times by “riding” with a punch, by how much will the force of impact be reduced? • Since the time of impact increases by 5 times, the force of impact will be reduced by 5 times.

15. BOUNCING • IMPULSES ARE GREATER WHEN AN OBJECT BOUNCES • The impulse required to bring an object to a stop and then to throw it back upward again is greater than the impulse required to merely bring the object to a stop. • When a martial artist breaks boards, • does their hand bounce? • Is impulse or momentum greater ? • Example : • The Pelton Wheel.

16. Calculating the Change of Momentum For the teddy bear For the bouncing ball

17. How Good Are the Bumpers? • In a crash test, a car of mass 1.5103 kg collides with a wall and rebounds as in figure. The initial and final velocities of the car are vi=-15 m/s and vf = 2.6 m/s, respectively. If the collision lasts for 0.15 s, find • (a) the impulse delivered to the car due to the collision • (b) the size and direction of the average force exerted on the car

18. How Good Are the Bumpers? • In a crash test, a car of mass 1.5103 kg collides with a wall and rebounds as in figure. The initial and final velocities of the car are vi=-15 m/s and vf = 2.6 m/s, respectively. If the collision lasts for 0.15 s, find • (a) the impulse delivered to the car due to the collision • (b) the size and direction of the average force exerted on the car

19. CONSERVATION OF MOMENTUM • To accelerate an object, a force must be applied. • The force or impulse on the object must come from outside the object. • EXAMPLES : The air in a basketball, sitting in a car and pushing on the dashboard or sitting in a boat and blowing on the sail don’t create movement. • Internal forces like these are balanced and cancel each other. • If no outside force is present, no change in momentum is possible.

20. The Law of Conservation of Momentum • In the absence of an external force, the momentum of a system remains unchanged. • This means that, when all of the forces are internal (for EXAMPLE: the nucleus of an atom undergoing . radioactive decay, . cars colliding, or . stars exploding the net momentum of the system before and after the event is the same.

21. Conservation of Momentum • In an isolated and closed system, the total momentum of the system remains constant in time. • Isolated system: no external forces • Closed system: no mass enters or leaves • The linear momentum of each colliding body may change • The total momentum P of the system cannot change.

22. Conservation of Momentum • Start from impulse-momentum theorem • Since • Then • So

23. QUESTIONS • 1. Newton’s second law states that if no net force is exerted on a system, no acceleration occurs. Does it follow that no change in momentum occurs? • No acceleration means that no change occurs in velocity of momentum. • 2. Newton’s 3rd law states that the forces exerted on a cannon and cannonball are equal and opposite. Does it follow that the impulse exerted on the cannon and cannonball are also equal and opposite? • Since the time interval and forces are equal and opposite, the impulses (F x t) are also equal and opposite.

24. COLLISIONS • ELASTIC COLLISIONS • INELASTIC COLLISIONS Momentum transfer from one Object to another . Is a Newton’s cradle like the one Pictured here, an example of an elastic or inelastic collision?

25. Problem Solving #1 • A 6 kg fish swimming at 1 m/sec swallows a 2 kg fish that is at rest. Find the velocity of the fish immediately after “lunch”. • net momentum before = net momentum after • (net mv)before = (net mv)after • (6 kg)(1 m/sec) + (2 kg)(0 m/sec) = (6 kg + 2 kg)(vafter) • 6 kg.m/sec = (8 kg)(vafter) • vafter = 6 kg.m/sec /8 kg • 8 kg • vafter = ¾ m/sec vafter =

26. Problem Solving #2 • Now the 6 kg fish swimming at 1 m/sec swallows a 2 kg fish that is swimming towards it at 2 m/sec. Find the velocity of the fish immediately after “lunch”. • net momentum before = net momentum after • (net mv)before = (net mv)after • (6 kg)(1 m/sec) + (2 kg)(-2 m/sec) = (6 kg + 2 kg)(vafter) • 6 kg.m/sec + -4 kg.m/sec = (8 kg)(vafter) • vafter = 2 kg.m/sec /8 kg • 8 kg • vafter = ¼ m/sec vafter =

27. Problem Solving #3 & #4 • Now the 6 kg fish swimming at 1 m/sec swallows a 2 kg fish that is swimming towards it at 3 m/sec. • (net mv)before = (net mv)after • (6 kg)(1 m/sec) + (2 kg)(-3 m/sec) = (6 kg + 2 kg)(vafter) • 6 kg.m/sec + -6 kg.m/sec = (8 kg)(vafter) • vafter = 0 m/sec • Now the 6 kg fish swimming at 1 m/sec swallows a 2 kg fish that is swimming towards it at 4 m/sec. • (net mv)before = (net mv)after • (6 kg)(1 m/sec) + (2 kg)(-4 m/sec) = (6 kg + 2 kg)(vafter) • 6 kg.m/sec + -8 kg.m/sec = (8 kg)(vafter) • vafter = -1/4 m/sec

28. The Archer • An archer stands at rest on frictionless ice and fires a 0.5-kg arrow horizontally at 50.0 m/s. The combined mass of the archer and bow is 60.0 kg. With what velocity does the archer move across the ice after firing the arrow?

29. Types of Collisions • Momentum is conserved in any collision • Inelastic collisions: rubber ball and hard ball • Kinetic energy is not conserved • Perfectly inelastic collisions occur when the objects stick together • Elastic collisions: billiard ball • both momentum and kinetic energy are conserved • Actual collisions • Most collisions fall between elastic and perfectly inelastic collisions

30. Simple Examples of Head-On Collisions (Energy and Momentum are Both Conserved) Collision between two objects of the same mass. One mass is at rest. Collision between two objects. One at rest initially has twice the mass. Collision between two objects. One not at rest initially has twice the mass.

31. Example of Non-Head-On Collisions (Energy and Momentum are Both Conserved) Collision between two objects of the same mass. One mass is at rest. If you vector add the total momentum after collision, you get the total momentum before collision.

32. Collisions Summary • In an elastic collision, both momentum and kinetic energy are conserved • In an inelastic collision, momentum is conserved but kinetic energy is not.Moreover, the objects do not stick together • In a perfectly inelastic collision, momentum is conserved, kinetic energy is not, and the two objects stick together after the collision, so their final velocities are the same • Elastic and perfectly inelastic collisions are limiting cases, most actual collisions fall in between these two types • Momentum is conserved in all collisions

33. More about Perfectly Inelastic Collisions • When two objects stick together after the collision, they have undergone a perfectly inelastic collision • Conservation of momentum • Kinetic energy is NOT conserved

34. An SUV Versus a Compact • An SUV with mass 1.80103 kg is travelling eastbound at +15.0 m/s, while a compact car with mass 9.00102 kg is travelling westbound at -15.0 m/s. The cars collide head-on, becoming entangled. Find the speed of the entangled cars after the collision. Find the change in the velocity of each car. Find the change in the kinetic energy of the system consisting of both cars.

35. An SUV Versus a Compact Find the speed of the entangled cars after the collision.

36. An SUV Versus a Compact Find the change in the velocity of each car.

37. An SUV Versus a Compact Find the change in the kinetic energy of the system consisting of both cars.

38. More About Elastic Collisions • Both momentum and kinetic energy are conserved • Typically have two unknowns • Momentum is a vector quantity • Direction is important • Be sure to have the correct signs • Solve the equations simultaneously

39. Elastic Collisions • A simpler equation can be used in place of the KE equation

40. Summary of Types of Collisions • In an elastic collision, both momentum and kinetic energy are conserved • In an inelastic collision, momentum is conserved but kinetic energy is not • In a perfectly inelastic collision, momentum is conserved, kinetic energy is not, and the two objects stick together after the collision, so their final velocities are the same

41. Problem Solving for 1D Collisions, 1 • Coordinates: Set up a coordinate axis and define the velocities with respect to this axis • It is convenient to make your axis coincide with one of the initial velocities • Diagram: In your sketch, draw all the velocity vectors and label the velocities and the masses

42. Problem Solving for 1D Collisions, 2 • Conservation of Momentum: Write a general expression for the total momentum of the system before and after the collision • Equate the two total momentum expressions • Fill in the known values

43. Problem Solving for 1D Collisions, 3 • Conservation of Energy: If the collision is elastic, write a second equation for conservation of KE, or the alternative equation • This only applies to perfectly elastic collisions • Solve: the resulting equations simultaneously

44. One-Dimension vs Two-Dimension

45. Two-Dimensional Collisions • For a general collision of two objects in two-dimensional space, the conservation of momentum principle implies that the total momentum of the system in each direction is conserved

46. Two-Dimensional Collisions • The momentum is conserved in all directions • Use subscripts for • Identifying the object • Indicating initial or final values • The velocity components • If the collision is elastic, use conservation of kinetic energy as a second equation • Remember, the simpler equation can only be used for one-dimensional situations

47. Glancing Collisions • The “after” velocities have x and y components • Momentum is conserved in the x direction and in the y direction • Apply conservation of momentum separately to each direction

48. Problem Solving for Two-Dimensional Collisions, 3 • Conservation of Energy: If the collision is elastic, write an expression for the total energy before and after the collision • Equate the two expressions • Fill in the known values • Solve the quadratic equations • Can’t be simplified

49. Problem Solving for Two-Dimensional Collisions, 4 • Solve for the unknown quantities • Solve the equations simultaneously • There will be two equations for inelastic collisions • There will be three equations for elastic collisions • Check to see if your answers are consistent with the mental and pictorial representations. Check to be sure your results are realistic

50. Collision at an Intersection • A car with mass 1.5×103 kg traveling east at a speed of 25 m/s collides at an intersection with a 2.5×103 kg van traveling north at a speed of 20 m/s. Find the magnitude and direction of the velocity of the wreckage after the collision, assuming that the vehicles undergo a perfectly inelastic collision and assuming that friction between the vehicles and the road can be neglected.