Momentum and Collisions

1
Chapter 6

• Momentum and Collisions

Conceptual questions 3,4,8,14 Quick quizzes
2,4,5,6 Problems 22,30,49,50
2
Momentum

• The linear momentum of an object of mass m moving
  with a velocity v is defined as the product of
  the mass and the velocity
• p m v
• SI Units are kg m / s
• Vector quantity, the direction of the momentum is
  the same as the velocitys
• Momentum components

3
Impulse

• In order to change the momentum of an object, a
  force must be applied
• The time rate of change of momentum of an object
  is equal to the net force acting on it
• Alternative form of the second Newton law

4
Impulse-Momentum Theorem

• When a single, constant force acts on the object
• F?t is defined as the impulse
• Vector quantity, the direction is the same as the
  direction of the force

• The impulse acting on the object is equal to the
  change in momentum of the object
• If the force is not constant, use the average
  force applied

5
Average Force in Impulse

• The average force can be thought of as the
  constant force that would give the same impulse
  to the object in the time interval as the actual
  time-varying force gives in the interval

6
Collision time

• The air bag increases the time of the collision
• The most important factor is the collision time
  or the time it takes the person to come to a rest
• Air bag will also absorb some of the energy from
  the body
• It will spread out the area of contact
• decreases the pressure
• helps prevent penetration wounds

7
Conservation of Momentum

• Momentum in an isolated system in which a
  collision occurs is conserved
• An isolated system will have not external forces
• When no external forces act on a system
  consisting of two objects that collide with each
  other, the total momentum of the system before
  the collision is equal to the total momentum of
  the system after the collision

8
Conservation of Momentum, cont.

• Mathematically
• Momentum is conserved for the system of objects
• The system includes all the objects interacting
  with each other
• Assumes only internal forces are acting during
  the collision

9
Quick quiz

• 6.2 A boy stands at one end of a floating raft
  that is stationary relative to the shore. He
  then walks to the opposite end of the raft, away
  from the shore. As a consequence, the raft
• remains stationary,
• moves away from the shore, or
• (c) moves toward the shore.

10
Problem 6-22
A 65.0-kg person throws a 0.045 0-kg snowball
forward with a ground speed of 30.0 m/s. A second
person, with a mass of 60.0 kg, catches the
snowball. Both people are on skates. The first
person is initially moving forward with a speed
of 2.50 m/s, and the second person is initially
at rest. What are the velocities of the two
people after the snowball is exchanged? Disregard
the friction between the skates and the ice.
11
Problem Solving for One -Dimensional Collisions

• 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
• In your sketch, draw all the velocity vectors
  with labels including all the given information

12
Types of Collisions

• Momentum is conserved in any collision
• Elastic collision
• Both momentum and kinetic energy are conserved
• Inelastic collisions
• Momentum is conserved but kinetic energy is not
  conserved
• Some of the kinetic energy is converted into
  other types of energy such as heat, sound, work
  to permanently deform an object
• Perfectly inelastic collisions occur when the
  objects stick together

13
Conceptual questions
3. In perfectly inelastic collision between two
objects, there are events in which all of the
original kinetic energy is transformed to forms
other than kinetic. Give an example of such an
event. 4. If two objects collide and one is
initially at rest, is it possible for both to be
at rest after the collision? Is it possible for
one of them to be at rest after the collision? 8.
If two particles have equal kinetic energies,
are their momenta necessarily equal?
14
Conceptual questions, cont.

• You are watching a movie about a superhero and
  notice that the superhero hovers in the air and
  throws a piano at some bad guys while remaining
  stationary in the air. What is wrong with this
  scenario?
• If you jumps from a table and land with your
  knees locked, why are you more likely to be hurt
  than if you land with your legs relaxed?

15
More About Elastic Collisions

• Both momentum and kinetic energy are conserved
• A simpler equation can be used in place of the
  second equation

16
Sketches for Collision Problems

• Draw before and after sketches
• Label each object
• include the direction of velocity
• keep track of subscripts

17
Problem Solving for One-Dimensional Collisions,
cont.

• Write the expressions for the momentum of each
  object before and after the collision
• Write an expression for the total momentum before
  and after the collision
• Remember the momentum of the system is what is
  conserved

18
Problem Solving for One-Dimensional Collisions,
final

• If the collision is inelastic, solve the momentum
  equation for the unknown
• Remember, KE is not conserved
• If the collision is elastic, you can use the KE
  equation (or the simplified one) to solve for two
  unknowns

19
More About Perfectly Inelastic Collisions

• When two objects stick together after the
  collision, they have undergone a perfectly
  inelastic collision
• Conservation of momentum becomes

20
Sketches for Perfectly Inelastic Collisions

• The objects stick together
• Include all the velocity directions
• The after collision combines the masses

21
Problem 6-49
Most of us know intuitively that in a head-on
collision between a large dump truck and a
subcompact car, you are better off being in the
truck than in the car. Why is this? Many people
imagine that the collision force exerted on the
car is much greater than that experienced by the
truck. To substantiate this view, they point out
that the car is crushed, whereas the truck is
only dented. This idea of unequal forces, of
course, is false. Newtons third law tells us
that both objects experience forces of the same
magnitude. The truck suffers less damage because
it is made of stronger metal. But what about the
two drivers? Do they experience the same forces?
To answer this question, suppose that each
vehicle is initially moving at 8.00 m/s and that
they undergo a perfectly inelastic head-on
collision. Each driver has mass 80.0 kg.
Including the drivers, the total vehicle masses
are 800 kg for the car and 4 000 kg for the
truck. If the collision time is 0.120 s, what
force does the seat belt exert on each driver?
22
Quick quiz 6.4
An object of mass m moves to the right with a
speed v. It collides head-on with an object of
mass 3m moving with speed v/3 in the opposite
direction. If the two objects stick together,
what is the speed of the combined object of mass
4m after the collision? (a) 0 (b) v/2 (c)
v (d) 2v
23
Problem 6-30
An 8.00-g bullet is fired into a 250-g block that
is initially at rest at the edge of a table of
height 1.00 m (Fig. P6.30). The bullet remains in
the block, and after the impact the block lands
2.00 m from the bottom of the table. Determine
the initial speed of the bullet.
24
Glancing Collisions

• For a general collision of two objects in
  three-dimensional space, the conservation of
  momentum principle implies that the total
  momentum of the system in each direction is
  conserved
• Use subscripts for identifying the object,
  initial and final, and components

25
Glancing Collisions

• The after velocities have x and y components
• Momentum is conserved in the x direction and in
  the y direction
• Apply separately to each direction

26
Problem Solving for Two-Dimensional Collisions

• Set up coordinate axes and define your velocities
  with respect to these axes
• It is convenient to choose the x axis to coincide
  with one of the initial velocities
• In your sketch, draw and label all the velocities
  and include all the given information
• Find px and py components for all momenta

27
Problem Solving for Two-Dimensional Collisions,
final

• Write expressions for total momentum before and
  after collision in the x direction and equate the
  two. Solve for unknowns.
• Repeat for the y-direction
• If the collision is inelastic, additional
  information is probably required
• If the collision is perfectly inelastic, the
  final velocities of the two objects is the same
• If the collision is elastic, use the KE equations
  to help solve for the unknowns

28
Quick quiz 6.5
Suppose you are blindfolded in a game room where
your friends are playing billiards. If you only
hear a clicking sound as one ball collides with
another, you would conclude that the collisions
are (a) elastic, (b) inelastic, (c) not enough
information is given.
29
Quick quiz 6.5
A skater is using very low friction roller
blades. A friend throws a Frisbee at her, along
the straight line along which she is coasting.
Describe each of the following events as an
elastic, an inelastic, or a perfectly inelastic
collision between the skater and the Frisbee
(a) She catches the Frisbee and holds it. (b)
She tries to catch the Frisbee but it bounces
off her hands and falls to the ground in front of
her. (c) She catches the Frisbee and
immediately throws it back with the same speed
(relative to the ground) to her friend.
30
Quick quiz 6.6
In a perfectly inelastic one-dimensional
collision between two objects, what condition
alone is necessary so that all of the original
kinetic energy of the system is gone after the
collision? (a) The objects must have momenta
with the same magnitude but opposite directions.
(b) The objects must have the same mass. (c)
The objects must have the same velocity. (d) The
objects must have the same speed, with velocity
vectors in opposite directions.
31
Rocket Propulsion

• The rocket is accelerated as a result of the
  thrust of the exhaust gases
• Momentum is conserved
• Kinetic Energy is increased (at the expense of
  the stored energy of the rocket fuel)

32
ve

• The initial mass of the rocket is M ?m
• M is the mass of the rocket
• m is the mass of the fuel
• The initial velocity of the rocket is v

• The rockets mass is M
• The mass of the fuel, ?m, has been ejected with
  speed ve with respect to the rocket
• The rockets speed has increased to v ?v

(M ?m)v M(v ?v) ?m(v ve)
M ?v ?m ve (-?M) ve
33
Rocket Propulsion, final

• Solution of the equations on the previous slide
  is the basic equation for rocket propulsion
• Mi is the initial mass of the rocket plus fuel
• Mf is the final mass of the rocket plus any
  remaining fuel
• The speed of the rocket is proportional to the
  exhaust speed

34
Thrust of a Rocket

• The thrust is the force exerted on the rocket by
  the ejected exhaust gases
• The instantaneous thrust is given by
• The thrust increases as the exhaust speed
  increases and as the burn rate (?M/?t) increases

35
Hint problem 6-57

• A 0.500-kg block is released from rest at the
  top of a frictionless track 2.50 m above the top
  of a table. It then collides elastically with a
  1.00-kg block that is initially at rest on the
  table, as shown in Figure P6.57.
• (a) Determine the velocities of the two blocks
  just after the collision.
• (b) How high up the track does the 0.500-kg
  block travel back after the collision?