Impulse and Momentum

1
Impulse and Momentum

• Chapter 7

2
Expectations

• After chapter 7, students will
• be able to define and calculate the impulse of a
  force.
• be able to define and calculate the momentum of a
  moving object.
• understand and apply the impulse-momentum
  theorem.
• understand the conditions in which the momentum
  of a system is conserved.

3
Expectations

• After chapter 7, students will
• distinguish between elastic and inelastic
  collisions.
• know what quantities are conserved in what sorts
  of collisions.
• be able to calculate the location and velocity of
  the center of mass of a system of objects.

4
Impulse and Momentum

• In the last chapter, we defined a force-related
  quantity (work) and a motion-related quantity
  (kinetic energy), and showed how they were
  linked
• This was the work-energy theorem.

5
Impulse and Momentum

• Now, we define a new force-related quantity the
  impulse of a force.
• The impulse is the product of the net force and
  the time over which it is exerted.
• It is a vector quantity, having the dimensions of
  force time. SI units Ns

6
Impulse and Momentum

• The net force F, acting on an object of mass m,
  produces an acceleration
• Substituting that into the definition of impulse
• Solve for at

7
Impulse and Momentum

• Substitute into the kinematic equation
• Multiply through by m

8
Impulse and Momentum

• Again, we have a relationship between a
  force-related quantity, I, and a motion-related
  quantity, mv.
• We define this quantity as linear momentum
• SI units kgm/s
• Note that linear momentum is a vector quantity.

9
Impulse and Momentum

• The above relationship is called the
  impulse-momentum theorem the impulse of a net
  force acting on an object is equal to the
  resulting change in its linear momentum.

10
Impulse and Momentum

• Bottom line a net force changes an objects
  state of motion. The work-energy theorem and the
  impulse-momentum theorem are two different
  descriptions of the same change.
• Which description is more useful to you depends
  on the situation what things you know, and what
  things you want to find out.

11
Systems of Objects

• A system of objects is any collection of objects.
• You define the system. You can say what objects
  are part of it, and what objects are outside of
  it.

12
Isolated Systems

• A system of objects is called isolated if the net
  external force acting on the system is zero.
• An external force is a force exerted on the
  system (any object in it) by an object that is
  not a part of the system.
• A force could be external or internal, depending
  on how the system is defined.

13
Conservation of Momentum

• The conservation of linear momentum principle
  tells us
• If a system is isolated, its total linear
  momentum does not change.
• Why not?

14
Conservation of Momentum

• The only force experienced by any part of an
  isolated system is an internal force a force
  exerted on one part of the system by another.
• Newtons third law tells us
• If objects A and B are both part of the system,
  then the net force on the system as a whole is

15
Conservation of Momentum

• If the net force is zero, its impulse is zero
• ... and if the impulse is zero, the change in
  (total) momentum is zero
• ... and the total linear momentum is conserved.

16
Conservation of Momentum

• We said the change in total (i.e., system)
  momentum was zero. Notice that the linear
  momentum of any particular object within the
  system will, in general, change due to internal
  forces. But the sum of the individual momenta of
  the objects in the system, added as the vectors
  that they are, does not change as long as the
  system is isolated.

17
Collisions

• If we define a system of two or more objects, it
  is usually because those objects are busy
  colliding with each other.
• In every collision among the objects in an
  isolated system, the total linear momentum of the
  system is conserved. We established that
  earlier, with the linear momentum conservation
  principle.

18
Elastic Collisions

• We now want to consider a special class of
  collisions elastic collisions. In an elastic
  collision between objects in an isolated system,
  the total kinetic energy of the system is also
  conserved.
• Why should that be?

19
Elastic Collisions

• The work-energy theorem tells us that the total
  or net work due to conservative forces does not
  change the total energy of the system.
• The gravitational potential energy of the system
  does not change unless the system experiences a
  net (external) gravitational force ... in which
  case, it is no longer an isolated system.

20
Elastic Collisions

• This means that, if only conservative forces do
  work within the system, its kinetic energy must
  be constant ... since we already know its
  potential energy is constant.
• What conservative forces do work within the
  system?
• Spring forces (as long as deformations are
  elastic).

21
Elastic Collisions

• If objects in the system are elastically deformed
  by the forces they exert on each other, the work
  done by those forces is stored as elastic
  potential energy. When the objects
  pre-collision shapes are restored, that elastic
  potential energy is re-converted into kinetic
  energy.
• So, the total kinetic energy of the system after
  the elastic collision is the same as before the
  collision.

22
Collision Summary

• If the system is isolated (no net external force
  acts on the system), momentum is conserved
• If only conservative forces do work in the
  collision, the collision is elastic, and kinetic
  energy is conserved

23
Collision Summary

• Collisions are not, in general, either perfectly
  elastic or perfectly inelastic. Those two
  conditions are idealizations that can give us
  insights into real collisions.

24
Center of Mass

• A system of several objects has its total mass
  distributed among those objects.
• When we consider the conservation of a systems
  total momentum, it will be useful to us to define
  a single location where the systems total mass
  is effectively located the center of mass.

25
Center of Mass

• Consider a system of two objects, both located on
  the X axis.
• If m1 m2 m,

26
Center of Mass

• Now, we let the objects in our system move
  (independently)

27
Center of Mass

• The motion takes place in a time Dt
• or, velocity of the
  center of mass

28
Center of Mass

• Notice that the center-of-mass velocity is the
  total
• momentum of the system, divided by its total
  mass.
• If the system is isolated,
• its total momentum is
• constant and so is its
• total mass. So, the
• center-of-mass velocity of an isolated system is
• constant, regardless of intra-system collisions.

for isolated systems!