Momentum

1
Momentum
2
Introduction

• We know that it is harder to get a more massive
  object moving from rest than a less massive
  object.
• This is the concept of inertia that we have
  already included in our world view.
• We also know intuitively that for two objects
  moving with the same speed, the more massive
  object would be the harder one to stop.
• For example, if a kayak and a cargo ship were
  both moving through the harbor at 5 kilometers
  per hour, you would clearly have an easier time
  stopping the kayak.

3
Introduction

• We need to add a new concept to our world view to
  address the question, How hard would it be to
  stop an object?
• We call this new concept momentum, and it depends
  on both the mass of the object and how fast it is
  moving.

The kayak and the ship have different momenta
even when they are moving at the same speed,
because of their different masses.
4
Linear Momentum

• The linear momentum of an object is defined as
  the product of its mass and its velocity.
• Momentum is a vector quantity that has the same
  direction as the velocity.
• Using the symbol p for momentum, we write the
  relationship as
• The adjective linear distinguishes this from
  another kind of momentum that we will discuss in
  Chapter 8.
• Unless there is a possibility of confusion, this
  adjective is usually omitted.

5
Linear Momentum

• There is no special unit for momentum as there is
  for force the momentum unit is simply that of
  mass times velocity (or speed)
• that is, kilogram-meter per second (kg m/s).
• An object may have a large momentum due to a
  large mass, a large velocity, or both.
• A slow battleship and a rocket-propelled race car
  have large momenta.

6
On the Bus

• Q Which has the greater momentum, an 18-wheeler
  parked at the curb or a Volkswagen rolling down a
  hill?
• A Because the 18-wheeler has zero velocity, its
  momentum is also zero. Therefore, the VW has the
  larger momentum as long as it is moving.

7
Linear Momentum

• The word momentum is often used in our everyday
  language in a much looser sense, but it is still
  roughly consistent with its meaning in the
  physics world view
• something with a lot of momentum is hard to stop.
  
• You have probably heard someone say, We dont
  want to lose our momentum!
• Coaches are particularly fond of this word.

8
Changing an Objects Momentum

• The momentum of an object changes if its velocity
  and/or its mass changes.
• We can obtain an expression for the amount of
  change by rewriting Newtons second law
  in a more general form.
• Actually, Newtons original formulation is closer
  to the new form.
• Newton realized that mass as well as velocity
  could change.
• His form of the second law says that the net
  force is equal to the change in the momentum
  divided by the time required to make this change

9
Changing an Objects Momentum

• If we now multiply both sides of this equation by
  the time interval ?t, we get an equation that
  tells us how to produce a change in momentum
• This relationship tells us that this change is
  produced by applying a net force to the object
  for a certain time interval.
• The interaction that changes an objects
  momentuma force acting for a time intervalis
  called impulse.
• Impulse is a vector quantity that has the same
  direction as the net force.

10
Changing an Objects Momentum

• Because impulse is a product of two things, there
  are many ways to produce a particular change in
  momentum.
• For example, two ways of changing an objects
  momentum by 10 kilogram-meters per second are to
  exert a net force of 5 newtons on the object for
  2 seconds or to exert 100 newtons for 0.1 second.
  
• They each produce an impulse of 10 newton-seconds
  (Ns) and therefore a momentum change of 10
  kilogram-meters per second.
• The units of impulse (newton-seconds) are
  equivalent to those of momentum (kilogram-meters
  per second).

11
On the Bus

• Q Which of the following will cause the larger
  change in the momentum of an object a force of 2
  newtons acting for 10 seconds or a force of 3
  newtons acting for 6 seconds?
• A The larger impulse causes the larger change in
  the momentum. The first force yields an impulse
  of
• (2 newtons)(10 seconds) 20 newton-seconds
• the second yields
• (3 newtons)(6 seconds) 18 newton-seconds.
• Therefore, the first impulse produces the larger
  change in momentum.

12
Changing an Objects Momentum

• Although the momentum change may be the same,
  certain effects depend on the particular
  combination of force and time.
• Suppose you had to jump from a second-story
  window. Would you prefer to jump onto a wooden or
  concrete surface?
• Intuitively, you would choose the wooden one.
• Our commonsense world view tells us that jumping
  onto a surface that gives is better.
• But why is this so?

13
Changing an Objects Momentum

• You undergo the same change in momentum with
  either surface your momentum changes from a high
  value just before you hit to zero afterward.
• The difference is in the time needed for the
  collision to occur.
• When a surface gives, the collision time is
  longer.
• Therefore, the average net force must be
  correspondingly smaller to produce the same
  impulse.

14
Changing an Objects Momentum

• Because our bones break when forces are large,
  the particular combination of force and time
  interval is important.
• For a given momentum change, a short collision
  time could cause large enough forces to break
  bones.

• You may break a leg landing on the concrete.
• On the other hand, the collision time with wood
  may be large enough to keep the forces in a huge
  momentum change from doing any damage.

The catcher is protected from the baseballs
momentum.
15
Changing an Objects Momentum

• This idea has many applications.
• Dashboards in cars are covered with foam rubber
  to increase the collision time during an
  accident.
• New cars are built with shock-absorbing bumpers
  to minimize damage to cars and with air bags to
  minimize injuries to passengers.
• The barrels of water or sand in front of highway
  median strips serve the same purpose.
• Stunt people are able to leap from amazing
  heights by falling onto large air bags that
  increase their collision times on landing.

16
Changing an Objects Momentum

• Volleyball players wear knee pads.
• Small pieces of polystyrene foam are used as
  packing material in shipping boxes to smooth out
  the bumpy rides.
• Even without a soft surface, we have learned how
  to increase the collision time when jumping.
• Instead of landing stiff-kneed, we bend our knees
  immediately on colliding with the ground.
• We are then brought to rest gradually rather than
  abruptly.

17
Conservation of Linear Momentum

• Imagine standing on a giant skateboard that is at
  rest. What is the total momentum of you and the
  skateboard?
• It must be zero because everything is at rest.
• Now suppose that you walk on the skateboard. What
  happens to the skateboard?
• When you walk in one direction, the skateboard
  moves in the other direction.

18
Conservation of Linear Momentum

• An analogous thing happens when you fire a rifle
  
• the bullet goes in one direction, and
• the rifle recoils in the opposite direction.
• These situations can be understood even though we
  dont know the values of the forcesand thus the
  impulsesinvolved.
• We start by assuming that there is no net
  external force to the objects.
• In particular, we assume that the frictional
  forces are negligible and that any other external
  forcesuch as gravityis balanced by other
  forces.

19
Conservation of Linear Momentum

• When you walk on the skateboard, there is an
  interaction.
• The force you exert on the skateboard is, by
  Newtons third law, equal and opposite to the
  force the skateboard exerts on you.
• The time intervals during which these forces act
  on you and the skateboard must be the same
  because there is no way that one can touch the
  other without also being touched.
• Because you and the skateboard each experience
  the same force for the same time interval, you
  must each experience the same-size impulse and
  therefore the same-size change in momentum.

20
Conservation of Linear Momentum

• But impulse and momentum are vectors, so their
  directions are important.
• Because the impulses are in opposite directions,
  the changes in the momenta are also in opposite
  directions.
• Thus, your momentum and that of the skateboard
  still add to zero.
• In other words, even though you and the
  skateboard are moving and, individually, have
  nonzero momenta, the total momentum of the system
  consisting of you and the skateboard remains
  zero.
• Notice that we arrived at this conclusion without
  considering the details of the forces involved.
  It is true for all forces between you and the
  skateboard.

21
On the Bus

• Q Suppose the skateboard has half your mass and
  you walk at a velocity of 1 meter per second to
  the left. Describe the motion of the skateboard.
• A The skateboard must have the same momentum but
  in the opposite direction. Because it has half
  the mass, its speed must be twice as much.
  Therefore, its velocity must be 2 meters per
  second to the right.

22
Conservation of Linear Momentum

• Because the changes in the momenta of the two
  objects are equal in size and opposite in
  direction, the value of the total momentum does
  not change.
• We say that the total momentum of the system is
  conserved.
• We can generalize these findings.
• Whenever any object is acted on by a force, there
  must be at least one other object involved.
• This other object may be in actual contact with
  the first, or it may be interacting at a distance
  of 150 million kilometers, but it is there.
• If we widen our consideration to include all of
  the interacting objects, we gain a new insight.

23
Conservation of Linear Momentum

• Consider the objects as a system.
• Whenever there is no net force acting on the
  system from the outside (that is, the system is
  isolated, or closed), the forces that are
  involved act only between the objects within the
  system.
• As a consequence of Newtons third law, the total
  momentum of the system remains constant.
• This generalization is known as the law of
  conservation of linear momentum.
• The total linear momentum of a system does not
  change if there is no net external force.

24
Conservation of Linear Momentum

• This means that if you add up all of the momenta
  now and leave for a while, when you return and
  add the momenta again, you will get the same
  vector even if the objects were bumping and
  crashing into each other while you were gone.
• In practice we apply the conservation of momentum
  to systems in which the net external force is
  zero or the effects of the forces can be
  neglected.

25
Conservation of Linear Momentum

• You experience conservation of momentum firsthand
  when you try to step from a small boat onto a
  dock.
• As you step toward the dock, the boat moves away
  from the dock, and you may fall into the water.
• Although the same effect occurs when we disembark
  from an ocean liner, the large mass of the ocean
  liner reduces the speed given it by our stepping
  off.
• A large mass requires a small change in velocity
  to undergo the same change in momentum.

26
Working it Out Momentum

• Lets calculate the recoil of a rifle.
• A 150-grain bullet for a .30-06 rifle has a mass
  m of 0.01 kg and a muzzle velocity v of 900 m/s
  (2000 mph).
• Therefore, the magnitude of the momentum p of the
  bullet is

27
Working it Out Momentum

• Because the total momentum of the bullet and
  rifle was initially zero, conservation of
  momentum requires that the rifle recoil with an
  equal momentum in the opposite direction.
• If the mass M of the rifle is 4.5 kg, the speed V
  of its recoil is given by
• If you do not hold the rifle snugly against your
  shoulder, the rifle will hit your shoulder at
  this speed (4.5 mph!) and hurt you.

28
On the Bus

• Q Why does holding the rifle snugly reduce the
  recoil effects?
• A Holding the rifle snugly increases the
  recoiling mass (your mass is now added to that of
  the rifle) and therefore reduces the recoil
  speed.

29
Conservation of Linear Momentum

• Although we dont notice it, the same effect
  occurs whenever we start to walk.
• Our momentum changes the momentum of something
  else must therefore change in the opposite
  direction.
• The something else is Earth.
• Because of its enormous mass, Earths speed need
  only change by an infinitesimal amount to acquire
  the necessary momentum change.

30
Collisions

• Interacting objects dont need to be initially at
  rest for conservation of momentum to be valid.
• Suppose a ball moving to the left with a certain
  momentum crashes head-on with an identical ball
  moving to the right with the same-size momentum.
• Before the collision, the two momenta are equal
  in size but opposite in direction, and because
  they are vectors, they add to zero.

31
Collisions

• After the collision the balls move apart with
  equal momenta in opposite directions.
• Because the masses of the balls are the same, the
  speeds after the collision are also the same.
• These speeds depend on the type of ball.
• The speeds may be almost as large as the original
  speeds in the case of billiard balls, quite a bit
  smaller in the case of lead balls, or even zero
  if the balls are made of soft putty and stick
  together.
• In all cases the two momenta are the same size
  and in opposite directions.
• The total momentum remains zero.

32
Collisions

• Consider the following example.
• A boxcar traveling at 10 meters per second
  approaches a string of four identical boxcars
  sitting stationary on the track.
• The moving boxcar collides and links with the
  stationary cars, and the five boxcars move off
  together along the track.
• What is the final speed of the five cars
  immediately after the collision?

33
Collisions

• Conservation of momentum tells us that the total
  momentum must be the same before and after the
  collision.
• Before the collision, one car is moving at 10
  meters per second.
• After the collision, five identical cars are
  moving with a common final speed.
• Because the amount of mass that is moving has
  increased by a factor of 5, the speed must
  decrease by a factor of 5.
• The cars will have a final speed of 2 meters per
  second.
• Notice that we did not have to know the mass of
  each boxcar, only that they all had the same
  mass.

34
Collisions

• We can use the conservation of momentum to
  measure the speed of fast-moving objects.
• For example, consider determining the speed of an
  arrow shot from a bow.
• We first choose a movable, massive targeta
  wooden block suspended by strings.

• Before the arrow hits the block, the total
  momentum of the system is equal to that of the
  arrow (the block is at rest).

35
Collisions

• After the arrow is embedded in the block, the two
  move with a smaller, more measurable speed.
• The final momentum of the block and arrow just
  after the collision is equal to the initial
  momentum of the arrow.

• Knowing the masses, we can determine the arrows
  initial speed.

36
Flawed Reasoning

• A question on the final exam asks, What do we
  mean when we claim that total momentum is
  conserved during a collision? The following two
  answers are given.
• Answer 1 Total momentum of the system stays the
  same before and after the collision.
• Answer 2 Total momentum of the system is zero
  before and after the collision.
• Which answer (if either) do you agree with?

37
Flawed Reasoning

• ANSWER Although we have considered several
  examples in which the total momentum of the
  system is zero, this is not the most general
  case.
• The momentum of a system can have any magnitude
  and any direction before the collision.
• If momentum is conserved, the momentum of the
  system always has the same magnitude and
  direction after the collision.
• Therefore, answer 1 is correct. This is a very
  powerful principle because of the word always.

38
Collisions

• Another example of a small, fast-moving object
  colliding with a much more massive object is
  graphically illustrated by one of your brave(?)
  authors, who lies down on a bed of nails.
• This in itself may seem like a remarkable feat.
  However, your author does not have to be a fakir
  with mystic powers, because he knows that the
  weight of his upper body is supported by 500
  nails so that each nail has to support only 0.4
  pound.
• It takes approximately 1 pound of force for the
  nail to break the skin.)
• Once on the bed of nails, he places a plate of
  nails on his chest and tops that off with a
  concrete block.

39
Collisions

• He then invites an assistant to smash the
  concrete block with a sledgehammer.
• This dramatic demonstration illustrates several
  ideas.

• The board on the chest spreads out the blow so
  that the force on any one part of the chest is
  small.
• Because it takes time for the hammer to break
  through the concrete block, the collision time is
  increased, and the force is therefore decreased
  even further.

• The concrete block is serving the same function
  as an air bag in a car (strangely enough).

40
Collisions

• Momentum is conserved in the collision, but the
  much larger mass of your author ensures that the
  velocity imparted to his body is much less than
  the velocity of the hammer.
• This means that his body is only slowly pushed
  down onto the nails, and the additional force
  that each nail must exert to stop his body is
  small.
• Therefore, your authors back is not perforated,
  and he lives to teach another day.

41
Investigating Accidents

• Accident investigators use conservation of
  momentum to reconstruct automobile accidents.
• Newtons laws cant be used to analyze the
  collision itself because we do not know the
  detailed forces involved.
• Conservation of linear momentum, however, tells
  us that regardless of the details of the crash,
  the total momentum of the two cars remains the
  same.
• The total momentum immediately before the crash
  must be equal to that immediately after the
  crash.
• Because the impact takes place over a very short
  time, we normally ignore frictional effects with
  the pavement and treat the collision as if there
  were no net external forces.

42
Investigating Accidents

• As an example, consider a rear-end collision.
• Assume that the front car was initially at rest
  and the two cars locked bumpers on impact.

• From an analysis of the length of the skid marks
  made after the collision and the type of surface,
  the total momentum of the two cars just after the
  collision can be calculated.
• We will see how to do this in Chapter 7 for now,
  assume we know their total momentum.

43
Investigating Accidents

• Because one car was stationary, the total
  momentum before the crash must have been due to
  the moving car.
• Knowing that the momentum is the product of mass
  and velocity (mv), we can compute the speed of
  the car just before the collision.
• We can thus determine whether the driver was
  speeding.

44
Working it Out Rear-Ended

• Lets use conservation of momentum to analyze
  this collision.
• For simplicity assume that each car has a mass of
  1000 kg (a metric ton) and that the cars traveled
  along a straight line.
• Further assume that we have determined that the
  speed of the two cars locked together was 10 m/s
  (about 22 mph) just after the crash.
• The total momentum after the crash was equal to
  the total mass of the two cars multiplied by
  their combined speed

45
Working it Out Rear-Ended

• But because momentum is conserved, this was also
  the value before the crash.
• Before the crash, however, only one car was
  moving.
• So if we divide this total momentum by the mass
  of the moving car, we obtain its speed
• The car was therefore traveling at 20 m/s (about
  45 mph) at the time of the accident.

46
On the Bus

• Q If the stationary car were not stationary but
  slowly rolling in the direction of the total
  momentum, how would the calculated speed of the
  other car change if the final momentum remained
  the same?
• A Because the rolling car accounts for part of
  the total momentum before the collision, the
  other car had less initial momentum and therefore
  a lower speed.

47
Investigating Accidents

• Assuming that the cars stick together after the
  collision simplifies the analysis but is not
  required.
• Conservation of momentum applies to all types of
  collisions.
• Even if the two cars do not stick together, the
  original velocity can be determined if the
  velocity of each car just after the collision can
  be determined.
• The cars do not even have to be going in the same
  initial direction.
• If the cars suffer a head-on collision, we must
  be careful to include the directions of the
  momenta, but the procedure of equating the total
  momenta before and after the accident remains the
  same.

48
Investigating Accidents

• Because momentum is a vector, this procedure can
  also be used in understanding two-dimensional
  collisions, such as when cars collide while
  traveling at right angles to each other.
• The total vector momentum must be conserved.

Studying physics in a billiard parlor improves
both your physics and your game.
49
Working it Out A General Collision

• Lets use the principle of conservation of
  momentum to solve a general collision problem.
• A 10-kg block is sliding to the right across a
  frictionless floor at 4 m/s.
• A 5-kg block is traveling left at 2 m/s such that
  it hits the other block head-on.
• After the collision, the 10-kg block is observed
  moving to the right at 1 m/s.
• Find the final speed of the 5-kg block.

50
Working it Out A General Collision

• To find the 5-kg blocks final velocity, we use
  the fact that momentum is conserved during the
  collision.
• We have enough information to find the initial
  momentum of the system
• Positive means to the right.
• After the collision, we know the momentum of the
  10-kg block
• Because the total must still equal 30 kgm/s, the
  final momentum of the 5-kg block must be 20
  kgm/s.
• This means that the 5-kg block is moving to the
  right after the collision with a speed of 4 m/s.

51
Airplanes, Balloons, and Rockets

• Conservation of momentum also applies to flight.
  If we look only at the airplane, momentum is
  certainly not conserved.
• It has zero momentum before takeoff, and its
  momentum changes many times during a flight.
• But if we consider the system of the airplane
  plus the atmosphere, momentum is conserved.

52
Airplanes, Balloons, and Rockets

• In the case of a propeller-driven airplane, the
  interaction occurs when the propeller pushes
  against the surrounding air molecules, increasing
  their momenta in the backward direction.
• This is accompanied by an equal change in the
  airplanes momentum in the forward direction

• If we could ignore the air resistance, the
  airplane would continually gain momentum in the
  forward direction.

53
Flawed Reasoning

• Two students are arguing about a collision
  between two gliders on an air track. Glider A
  hits glider B, which is twice as large and
  initially stationary.
• Jose I think that glider B will have the
  largest final speed when glider As final speed
  is zero. In this case, glider A gives all of its
  momentum to glider B.
• Shaq You are forgetting that momentum is a
  vector. If glider A bounces backward during the
  collision, it experiences a greater change in
  momentum than if it stops. Glider B must always
  experience the same change in momentum (but in
  the opposite direction) as glider A, so it would
  have a faster final speed in this case.
• With which student (if either) do you agree?

54
Flawed Reasoning

• ANSWER Shaq was paying attention in class. Anyone
  who has credit cards knows that it is possible to
  lose more than everything you have.
• If glider A is initially moving at 3 meters per
  second in the positive direction and stops, its
  change in velocity is 3 meters per second.
• If, on the other hand, the same glider bounces
  back with a final velocity of 2 meters per
  second, the change in its velocity is 5 meters
  per second.
• Because the change in momentum is just the mass
  times the change in velocity, bouncing results in
  the greater change of momentum.

55
On the Bus

• Q Why doesnt the airplane continually gain
  momentum?
• A As the airplane pushes its way through the
  air, it hits air molecules, giving them impulses
  in the forward direction. This produces impulses
  on the airplane in the backward direction. In
  straight, level flight at a constant velocity,
  the two effects cancel.

56
Airplanes, Balloons, and Rockets

• Release an inflated balloon, and it takes off
  across the room.
• Is this similar to the propeller-driven airplane?
  
• No, because the molecules in the atmosphere are
  not necessary.
• The air molecules in the balloon rush out,
  acquiring a change in momentum toward the rear.
• This is accompanied by an equal change in
  momentum of the balloon in the forward direction.
  
• The air molecules do not need to push on
  anything the balloon can fly through a vacuum.

57
Airplanes, Balloons, and Rockets

• This is also true of rockets and explains why
  they can be used in spaceflight.
• Rockets acquire changes in momentum in the
  forward direction by expelling gases at very high
  velocities in the backward direction.
• By choosing the direction of the expelled gases,
  the resulting momentum changes can also be used
  to change the direction of the rocket.

• An interesting classroom demonstration of this is
  often done using a modified fire extinguisher as
  the source of the high-velocity gas.

58
Airplanes, Balloons, and Rockets

• Jet airplanes lie somewhere between
  propeller-driven airplanes and rockets.
• Jet engines take in air from the atmosphere, heat
  it to high temperatures, and then expel it at
  high speed out the back of the engine.
• The fast-moving gases impart a change in momentum
  to the airplane as they leave the engine.
• Although the gases do not push on the atmosphere,
  jet engines require the atmosphere as a source of
  oxygen for combustion.

59
Summary

• The momentum of an object changes if its velocity
  or its mass changes.
• This change is produced by an impulse, a net
  force acting on the object for a certain time
  FDt.
• Impulse is a vector quantity with the same
  direction as the force
• this is also the direction of the change in
  momentum.
• There are many ways of producing a particular
  change in momentum by changing the strength of
  the force and the time interval during which it
  acts.

60
Summary

• The momentum of a system is the vector sum of all
  the momenta of the systems particles.
• Assuming that no net external force is acting on
  the system, the total momentum does not change.
• This generalization is known as the law of
  conservation of linear momentum.
• Conservation of momentum applies to many systems,
  such as balloons and billiard balls.