Momentum

1
Momentum

• Can be defined as inertia in motion
• (or by Newton an objects quantity of
  motion)
• Equation p m v
• units kg m/s
• So if the objects at rest, then its p 0 no
  matter how massive it is.
• Since momentum is based on an objects velocity,
  which is a vector quantity, it too is an vector
  quantity direction matters again!

2
Change in Momentum

• ?p ? p of an object!
• ?p pf pi where each term can have different
  values.
• Since ?p most commonly from a ?v,
• ?p mvf mvi
• ?p m (vf vi ) m?v, but dont use that
  in math solns!
• But it can also be from a ?m
• Ex full vs empty salt truck
• rocket ship burning fuel

3

• True or False?
• If an objects ?p 0, then its p 0 as well.
• False
• Ex Any moving object with a constant velocity
• If an objects p 0, then its ?p 0 as well.
• also False, although not as often
• Only if p 0 only lasts for an instant during
  the time when were determining ?p
• Ex a ball at the top of a free fall climb

4
The Cause of a Change in Momentum

• ?p is caused by a net force applied for a period
  of time
• This is called impulse (J)
• equations
• J ?p SF?t m (vf vi)
• units N s kg m/s
• (kg m/s2) s kg m/s
• Notice how net force can be defined in terms of
  p?
• SF ?p/?t or net F rate of ?p
• This is actually how Newton originally stated his
  2nd law in his book, Philosophiæ Naturalis
  Principia Mathematica aka

5
Mathematical Principles of Natural Philosophy

• or often simply referred to as the Principia
• a work in three books by Isaac Newton
• first published in Latin in July 1687.
• Lex II Mutationem motus proportionalem
• esse vi motrici impressae, et fieri secundum
• lineam rectam qua vis illa imprimitur.
• Translated by Motte 1729 as Law II The
  alteration of motion is ever proportional to the
  motive force impress'd and is made in the
  direction of the right line in which that force
  is impress'd.
• So, according to modern ideas of how Newton was
  using his terminology, Law II The change of
  momentum of a body is proportional to the impulse
  impressed on the body, and happens along the
  straight line on which that impulse is impressed.

6
The Cause of a Change in Momentum

• During a collision, net force can be assumed to
  be due to the forces between the interacting
  objects.
• Any other forces, like gravity or friction, are
  typically small compared to those between the
  colliding objects.
• So SF in SF?t, can just be F, often its
  actually avg F
• ?t the time it takes the collision to take
  place
• is often very short watch
• Classic
• F vs ?t
• graphs
• for a
• typical
• collision

7
Impending Collisions

• Notice, since ?p m (vf vi) J F ?t,
• If an object is destined to undergo a particular
  ?p (usually because it is going to crash or
  collide with something else) then force and time
  are inversely proportional to each other for that
  situation.
• Since most often it is a lot of force that
  causes damage to things, we often want to
  minimize that force, so we try to extend the
  amount of time the change in momentum (collision)
  takes place in because,
• The longer the time an object takes to change
  momentum, the less force will be needed,
  therefore the less damage to the object.
• Ex hard floor vs carpet
• cushioning a catch
• landing bent legged vs stiff

8

• More examples Run-away truck ramps

9

• More examples Air bags in vehicles
• BUT sometimes we want to damage the object, so
  a lot of force for a short time is ok
• Ex hammering a nail into a wall

10
The Significance of Bouncing

• When an object bounces, not only did something
  have to get the object stopped from its original
  motion, but then it also needed to get it moving
  again from rest in the opposite direction
• This requires a greater ?p or impulse than to
  simply stop the object from moving and since a
  bounce off often takes no more time than just
  stopping, it follows that a larger force is
  present
• Examples
• Pelton paddle wheel
• Rubber bumpers on cars??
• (like those on Grande Prix)
• karate chop that doesnt work

11
The Law of Conservation of Momentum

• The momentum of any isolated system remains
  constant.
• system refers to the objects youve chosen to be
  included - if they interact, its with INTERNAL
  forces
• isolated a system that has no net force being
  applied from objects outside the defined system
  - called EXTERNAL forces
• 2 Ways Con of p is Easily Seen
• if one object loses p,
• then another one must gain p
• Ex billiard balls during a game of pool
• Newtons Cradle
• air track

12
Figure 7-5Connecting Train Cars
What can we say about the CG of the system
before? After?
13
Figure 7-34 Bumper Car Rear End Collision
CG of the system before? After?

• vA 3.62 m/s vB 4.42 m/s
• Math is not doable in head, but note
• Slower one gains speed,
• Faster one loses speed,
• vA - vB - (vA - vB )
• 0.8 m/s - (-0.8 m/s) Note always true
  for 1D elastic

14
The Law of Conservation of Momentum

• The momentum of any isolated system remains
  constant. 2 Ways Con of p is Easily Seen
• if one object if one object starts moving one
  way, then another will move the opposite way with
  p
• Ex 2 students face off on skateboards

15
Example of Conservation of MomentumRocket
Propulsion
Recall by 3rd law, Fr on g -Fg on r
Now if we ?t ?t Then we get ?pg
bc r -?pr bc g con of p!
What can we say about the CG of the system
before? After?
16
Example of Conservation of MomentumRecoil of a
Fired Gun
CG of the system before? After?
17
Figure 7-3 Momentum is conserved in a collision
of two balls, labeled A and B.
Where is the CG of the system before?
After?
18
Forces on the balls during the collision of Fig.
7-3.

• FBonA FAonB
• These are the A/R forces from Newtons 3rd Law
• To find the net internal force of an isolated
  system, they will be added together!
• And will, therefore, cancel
• So the SFint 0 for any isolated system

19
Conservation of Momentum

• When it comes to defining your system, you get to
  pick the objects included in the system
• If theyre not in, and they apply a force, its
  an external force, so then its no longer an
  isolated system, so we cant expect p to be
  conserved.
• So then ?psys ? 0, and
• Impulse (J) ?psys Fext ?t m (vf vi)
• Ex 1. push on car from outside of it
• 2. drop ball it accelerates to ground
• 3. any interaction /problem from 7.1 7.3
• These are NOT exceptions to the law of
  conservation of momentum, we just arent
  satisfying the requirements of the law!

20
Conservation of Momentum

• When it comes to defining your system, you get
  to pick the objects included in the system
• If theyre in and they apply a force, its an
  internal force, so ?psys 0 for the system as a
  whole, even though objects inside may be changing
  their individual ps
• Ex 1. 2 students face off on skateboards
• 2. push on car from inside it
• 3. push car from outside, but include earth
• 4. watch ball drop, but include earth
• Does the Earth really gain - momentum?
• Yes but too small to measure
• No since it probably isnt even a net force

21

• The math for the law of conservation of p
• for an isolated system pisys pfsys
• pi1 pi2 pf1 pf2
• m1vi1 m2vi2 m1vf1 m2vf2
• m1
• m2
• vi1
• vi2
• vf1
• vf2

• Back to vectors big time!! Use directions!
• ID the given carefully use sketch with dir key
• Instead of subscripts 1 2, use letters that
  represent specific objects in the problem
• If something starts from or goes to rest, then
  that entire term 0
• If the objects are stuck together, then they
  have vs so you can pull it out as a common
  factor use (m1 m2) vi , not m1vi1 m2vi2

22
Types of Interactions

• Elastic Collisions kinetic energy is conserved
  before and after the collision
• But not during the collision KE ? PEe, until
  minimum separation distance is reached
• So then theres no heat loss, no permanent
  deformation, no sound created!
• Ex atomic molecular collisions
• But, in the macro world, only occurs as an ideal
  
• since really theres always some energy lost to
  heat, so true elastic collisions dont really
  exist
• but in Physicsland, we have close
  approximations
• Ex billiard balls in a game of pool
• steel balls on Newtons cradle
• spring bumpers on gliders on air track

23
Types of Interactions

• Inelastic Interactions where KE is not
  conserved
• Inelastic Collisions KEi gt KEf
• Ex minor car accident, bouncing ball, any hit in
  sports, fired bullet passes straight thru object
• Inelastic Explosions KEi lt KEf
• Ex firecracker, bomb (spring bug toy- not
  isolated)
• Completely Inelastic Collision where not only
  is KE not conserved, but the maximum lost during
  the collision is how much stays lost even after
  its over.
• When the objects entangle / stick together as a
  result of the collision maximum permanent
  deformation
• Ex serious car accident, fired bullet lodges
  into object, train cars hooking up (gliders w/
  magnets)
• Note p is conserved throughout all types of
  interactions!

24
Figure 7-13Elastic and Inelastic Collisions
(e) If completely inelastic, theyd have stuck
together maybe still moving, maybe not
25
The Math of Different Types of Interactions

• Any Isolated Interaction (elastic or inelastic)
• If only 1 unknown only need 1 eqn
• Use Law of Con of p pisys pfsys
• Elastic Collisions
• If 2 unknowns need 2 eqns
• Use Law of Con of p pisys pfsys
• Use Con of KE Ki Kf
• Head-on
• Use Law of Con of p pisys pfsys
• Use vi1 vi2 vf2 vf1 (see derivation
  p.176)

26
The Math of Different Types of Interactions

• Inelastic Collisions classic ex ballistic
  pendulum
• Must watch because often outside forces act at
  least for a portion of the entire situation, and
  if so,
• then ?psys ? 0 ? p is not conserved in that
  portion
• For the collison portion, its isolated, so
• Use Law of Con of p pisys pfsys
• For the stages where there are external forces,
  then try to conserve energy
• In general, use L of C of E Emechi WNC
  Emechf
• If external forces are conservative forces,
• then WNC 0
• If external forces are nonconservative forces,
• then usually Emechf 0

27
Ballistic Pendulum Set Up

• Stage I isolated system
• bullet moving toward stationary block
• Stage II bullet imbeds in block, moving
  together
• still isolated, but total inelastic collision
• Stage III gravity transforms all KE into PEg
• no longer isolated,
• but only conservative forces acting
• But what if block was not a pendulum, but
  attached to a spring that either extended or
  compressed after catching the bullet?
• Or slid to rest along a surface after catching
  the bullet?

28
Momentum Conservation in 2 or 3 Dimensions

• Individual momentum vectors of the various
  objects must be used
• then combine the momentum vectors by vector
  addition
• recall tip to tail or parallelogram method?
• then apply rt ? trig or law of sine/cosine to
  solve?
• and still, pisys pfsys
• Try sample problem on handout
  

What can we say about the CG of the system
before? After?