MOMENTUM

1
MOMENTUM COLLISIONS
2
Linear Momentum

• Moving objects have momentum
• Vector quantity
• Points in the same direction as the velocity
  vector
• Momentum
• Equals the product of an objects mass and
  velocity
• Proportional to mass and velocity
• p mv
• p momentum (kg m/s)
• m mass (kg)
• v velocity (m/s)

3
TAXI PROBLEM
What is the taxi cabs momentum? Mass of the
taxi 53 kg Velocity of the taxi 1.2
m/s Answer p mv p (53 kg)(1.2 m/s) p
63.6 kg m/s to the left
p 63.6 kg m/s
v 1.2 m/s
4
Momentum Newtons 2nd Law

• Newtons 2nd Law
• SF ma m(?v/?t)
• SF m(?v/?t)
• Momentum
• p mv ? m p/v
• SF (p/v)(?v/?t) ? SF ?p/?t

5
Impulse change In momentum

• If the momentum of an object changes, either
  mass, velocity, or both change
• If mass remains the same ? than velocity changes
  ? acceleration occurs
• What produces an acceleration?
• FORCE
• Greater the force acting on the object ? greater
  its change in velocity ? greater its change in
  momentum

6
Impulse change in momentum

• How long the force acts is also important
• Stalled car
• Apply a force over a brief amount of time ?
  produce a change in momentum
• Apply the same force over an extended period of
  time ? produce a greater change in the cars
  momentum
• A force suspended for a long time produces more
  change in momentum than does the same force
  applied briefly
• Both force and time are important in changing
  momentum

7
Impulse change in momentum

• IMPULSE (J) ?p pf pi mvf mvi
• J Favg ?t
• Favg?t m?v
• Favg m?v/?t
• Impulse (J) Change in momentum
• Impulse is also the product of the average force
  and the time during which the force is applied.
• Vector quantity
• Units kg m/s

8
Impulse problem

• A long jumper's speed just before landing is 7.8
  m/s. What is the impulse of her landing? (mass
  68 kg)
• J ?p
• J pf - pi
• J mvf mvi
• J 0 - (68kg)(7.8m/s)
• J -530 kg m/s
• Negative sign indicates that the direction of
  the impulse is opposite to her direction of motion

9
Impulsive force

• Baseball player swings a bat and hits the ball,
  the duration of the collision can be as short as
  1/1000th of a second and the force averages in
  the thousands of newtons
• The brief but large force the bat exerts on the
  ball Impulsive force

10
Kinetic Books

• View Kinetic books section 8.4- Physics at play
  Hitting a baseball
• BASEBALL PROBLEM
• The ball arrives at 40 m/s and leaves at 49 m/s
  in the opposite direction. The contact time is
  5.010-4 s. What is the average force on the
  ball?
• J ?p Favg ?t m?v
• Favg ?t m?v
• Favg?t m?v
• Favg m?v/?t
• Favg (0.14kg)(49 (-40)m/s)/5.010-4 s
• Favg 2.5104 N

11
Impulse change in momentum

• Case 1 Increasing momentum
• To increase the momentum of an object ? apply the
  greatest force possible for as long as possible
• Golfer teeing off and a baseball player trying
  for a home run
• Swing as hard as possible (large force)
• Follow through with their swing (increase in
  time)

12
Impulse change in momentum

• Case 2 Decreasing momentum
• You are in a car that is out of control ? Do you
  want to hit a cement wall or haystack?
• In either case, your momentum is decreased by the
  same impulse
• But, the same impulse does not mean the same
  amount of force or the same amount of time ?
  rather it means the same PRODUCT of force and
  time

13
Impulse change momentum

• Case 2 continued Decreasing momentum
• Hit the haystack ? Extend the impact time
• Change in momentum occurs over a long time ?
  Small impact force
• mv Ft
• Hit the cement wall
• Change in momentum occurs over a short time ?
  Large impact force
• mv Ft

14
Changing Momentum Scenario 1

• If you want to decrease a large momentum, you
  can have the force applied for a longer time
• If the change in momentum occurs over a
  long time ? Force of impact is small
• Examples
• Air bags in cars.
• Crash test video

FDt
15
Changing Momentum Scenario 2

• If the change in momentum occurs over a short
  time, the force of impact is large.
• Karate link
• Boxing video

FDt
16
Impulse change in momentum

• QUESTION
• When a glass falls, will the impulse be less if
  it lands on a carpet than if it lands on a hard
  floor?
• NO? Impulse is the same for either surface
  because the change in momentum is the same
• Carpet More time is available for the change in
  momentum ? smaller force for the impulse
• Hard floor Less time is available for the change
  in momentum (due to less give) ? larger force
  for the impulse

17
Conservation of momentum

• Conservation of momentum
• Occurs when there are no net external force(s)
  acting on the system
• Result ? Total momentum of an isolated system is
  constant
• Momentum before Momentum after
• Playing pool example
• Kinetic books 8.6

18
Conservation of momentum

• Momentum
• p mv
• Conservation of momentum
• Momentum before Momentum after
• pi1 pi2 pin pf1 pf2 pfn
• pi1, pi2, , pin initial momenta
• pf1, pf2, , pfn final momenta
• m1vi1 m2vi2 m1vf1 m2vf2
• m1, m2 masses of objects
• vi1, vi2 initial velocities
• vf1, vf2 final velocities

19
Conservation of momentum

• A 55.0 kg astronaut is stationary in the
  spaceships reference frame. She wants to move at
  0.500 m/s to the left. She is holding a 4.00 kg
  bag of dehydrated astronaut chow. At what
  velocity must she throw the bag to achieve her
  desired velocity? (Assume the positive direction
  is to the right.)

20
solution

• VARIABLES
• Mass of astronaut ma 55 kg
• Mass of bag mb 4 kg
• Initial velocity of astronaut via 0 m/s
• Initial velocity of bag vib 0 m/s
• Final velocity of astronaut vfa -0.5 m/s
• Final velocity of bag vfb ?
• EQUATION
• m1vi1 m2vi2 m1vf1 m2vf2
• mavia mbvib mavfa mbvfb
• 0 mavfa mbvfb
• Vfb - (mavfa / mb)
• Vfb - ((55kg)(-0.5m/s))/(4kg) 6.875 m/s

21
collisions

• Collision of objects ? Demonstrates the
  conservation of momentum
• Whenever objects collide in the absence of
  external forces
• net momentumbefore collision net momentumafter
  collision

22
collisions

• Momentum is conserved in ALL TYPES of collisions
• Elastic Collisions
• Objects collide without being permanently
  deformed and without generating heat
• Inelastic Collisions
• Colliding objects become distorted (tangled or
  coupled together) and generate heat

23
collisions

• Problem
• Consider a 6-kg fish that swims toward and
  swallows a 2-kg fish that is at rest. If the
  larger fish swims at 1 m/s, what is its velocity
  immediately after lunch?
• net momentumbefore collision net momentumafter
  collision
• (net mv)before (net mv)after
• (6kg)(1m/s) (2kg)(0) (6kg 2kg)(vafter)
• vafter ¾ m/s

24
collisions

• Problem
• Consider a 6-kg fish that swims toward and
  swallows a 2-kg fish that is moving towards the
  larger fish at 2 m/s. If the larger fish swims at
  1 m/s, what is its velocity immediately after
  lunch?
• net momentumbefore collision net momentumafter
  collision
• (net mv)before (net mv)after
• (6kg)(1m/s) (2kg)(-2m/s) (6kg 2kg)(vafter)
• vafter 1/4 m/s

25
collisions

• Perfectly Elastic collisions
• Not common in the everyday world
• Some heat is generated during collisions
• Drop a ball and after it bounces from the floor,
  both the ball and the floor are a bit warmer
• At the microscopic level ? perfectly elastic
  collisions are common
• Electrically charged particles bounce off one
  another without generating heat

26

Examples of Perfectly ELASTIC Collisions

• Electron scattering
• Hard spheres (Pool balls)

27
collisions

• Elastic collision
• Kinetic energy is conserved
• KE before KE after
• KE 1/2mv2
• Momentum is conserved in any collision ? Elastic
  or inelastic

28
ELASTIC collisions in 1-dimension

• Conservation of Kinetic Energy
• Conservation of Momentum
• Rearrange both equations and divide

29
Elastic collisions

• Final velocities in Head-On Two-Body Elastic
  Collisions (v2i 0 m/s)

30

Examples of Perfectly INELASTIC Collisions

• Catching a baseball Video
• Football tackle
• Cars colliding and sticking
• Bat eating an insect

31
collisions

• Inelastic collision
• Kinetic energy is NOT conserved
• KE before ? KE after
• Momentum is conserved in any collision ? Elastic
  or inelastic

32
Perfectly INELASTIC collisionsin 1-dimension

• Final velocities are the same

33
Problem
A 5879-lb (2665 kg) Cadillac Escalade going 35
mph smashes into a 2342-lb (1061 kg) Honda Civic
also moving at 35 mph (15.64 m/s) in the opposite
direction. The cars collide and stick.
a) What is the final velocity of the two
vehicles?

• m1v1i m2v2i (m1 m2)vf
• (2665kg)(15.64m/s) (1061kg)(-15.64m/s) (2665
  1061kg)vf
• vf 6.73 m/s 15.1 mph

34
Collisions

• Momentum is always conserved in a collision
• Collision video
• Classification of collisions
• ELASTIC
• Both energy momentum are conserved
• INELASTIC
• Momentum conserved, not energy
• Perfectly inelastic -gt objects stick
• Lost energy goes to heat

35
Center of mass

• Average location of mass
• An object can be treated as though all its mass
  were located at this point
• For a symmetric object made from a uniformly
  distributed material, the center of mass is the
  same as its geometric center

36
Center of mass

• Equation
• xcm m1x1 m2x2 mnxn / m1 m2 mn
• xCM x position of center of mass
• mi mass of object i
• xi x position of object i

37
Center of mass

• View section 8.20 in Kinetic books
• Specifically example 1- Center of mass problem

38
Center of Mass

• Video
• Balancing Activity video demo

39
Dont use following slides???
40
Conservation of momentum

• Key Facts
• Newtons 2nd Law (F ma)
• To accelerate an object ? Net force must be
  applied
• To change the momentum of an object ? exert an
  impulse on it
• The momentum of a system cannot change unless it
  is acted on by external forces

41
Conservation of momentum

• Law of Conservation of Momentum
• In the absence of an external force, the momentum
  of a system remains unchanged
• Examples in which the net momentum is the same
  before and after the event
• Radioactive decay
• Cars colliding
• Stars exploding

42
Conservation of Momentum
mv(initial) mv(final) An astronaut of mass
80 kg pushes away from a space station by
throwing a 0.75-kg wrench which moves with a
velocity of 24 m/s relative to the original frame
of the astronaut. What is the astronauts recoil
speed?
(0.75kg)(24m/s) 80kg(v) v 0.225 m/s
43
Conservation of momentum

• Question
• Newtons 2nd law states that if no net force is
  exerted on a system, no acceleration occurs. Does
  it follow that no change in momentum occurs?
• Yes, because no acceleration (a ?v/t) ? means
  no change in velocity ? and no change in momentum
  (p m?v)
• Also, no net force means ? no net impulse (J
  Ft) ? J ?p ? no change in momentum

44
Conservation of momentum

• Question
• Newtons 3rd law states that the force a rifle
  exerts on a bullet is equal and opposite to the
  force the bullet exerts on the rifle. Does is
  follow that the impulse the rifle exerts on the
  bullet is equal and opposite to the impulse the
  bullet exerts on the rifle?
• Yes, because the rifle acts on the bullet and
  bullet reacts on the rifle during the same time
  interval
• Since time is equal and force is equal and
  opposite for both ? Impulse, Ft, is also equal
  and opposite for both (Impulse vector quantity
  and can be canceled)

45
Conservation of momentum

• The law of conservation of momentum can be
  derived from Newtons 2nd and 3rd laws
• Newtons 2nd law ? F ma
• Newtons 3rd law ? Forces are equal but opposite
• Refer to Kinetic Books- 8.7 For step-by-step
  derivation

46
collisions

• Collisions
• Momentum- Useful concept when applied to
  collisions
• In a collision, two or more objects exert forces
  on each other for a brief instant of time, and
  these forces are significantly greater than any
  other forces they may experience during the
  collision

47
Problem
A proton (mp1.67x10-27 kg) elastically collides
with a target proton which then moves straight
forward. If the initial velocity of the
projectile proton is 3.0x106 m/s, and the target
proton bounces forward, what are a) The final
velocity of the projectile proton? b) The final
velocity of the target proton?
0.0 m/s 3.0 x 106 m/s
48
Elastic collision in 1-dimension

• Final equations for head-on elastic collision
• Relative velocity changes sign
• Equivalent to Conservation of Energy

49
Problem
An proton (mp1.67x10-27 kg) elastically collides
with a target deuteron (mD2mp) which then moves
straight forward. If the initial velocity of the
projectile proton is 3.0x106 m/s, and the target
deuteron bounces forward, what are a) The final
velocity of the projectile proton? b) The final
velocity of the target deuteron?
vp -1.0 x 106 m/s vd 2.0 x 106 m/s Head-on
collisions with heavier objects always lead to
reflections