Download Momentum and Its Conservation

Momentum and Its
Conservation
Chapter 9
Properties of a System
Up until now, we have looked at the
forces, motion and properties of single
isolated objects.
 In the following chapters (9,10,11), we
will look at the properties of a system of
objects and how they interact.
 Most of the time we will be looking at
before and after scenarios

Momentum
 Momentum
measures the strength of
an object’s motion.
 Momentum (p) depends on an
object’s mass and its velocity.
 More mass = more momentum
 More velocity = more momentum
Momentum
 Momentum
p
(p) = mass x velocity
= mv
 Momentum is a vector quantity!
Momentum acts in the same direction
as the velocity.
 Unit = kg m/s
– Do not confuse with a Newton
Impulse and Changing
Momentum
 To
change the momentum a force is
required but is not the only factor in
changing momentum
 To change the momentum an
Impulse is required.
 Impulse is the product of the Applied
Force and the Time Interval in which
the force acts.
Impulse and Changing
Momentum
 Impulse
= FDt
 Impulse is a vector quantity (it is in
the same direction as the force).
 Unit = Ns or kg m/s
(same as momentum)
 How can we have a Large Impulse?
Two Ways to Have a
Large Impulse
LARGE FORCE over a small time
interval
2. Small Force over a LARGE TIME
INTERVAL
1.
Impulse From a F vs. t Graph
We can find the Impulse from a Force
vs. Time Graph.
 The Impulse will equal the Area under
the curve of a F vs. t graph.
 If the Area is a rectangle use A = lw
 If the Area is a triangle use A = ½ bh

Impulse-Momentum Theorem
 An
Impulse causes a change in
Momentum. (If the mass remains
constant then the velocity of the
object must change.)
 FDt = Dp = mDv
 Note: Another way of expressing
Newton’s 2nd Law
Angular Momentum
Objects traveling with a linear
momentum (p) are affected by the mass
of the object and the velocity
 Rotating objects have an angular
momentum (L). An object’s angular
Momentum depends on how the mass
is distributed around the axis and its
angular speed!

Angular Momentum
L=Iw
 Similar to p = mv
 The Impulse Momentum Theorem also
applies to Angular Momentum (L).

 tDt = DL = Lf - Li
Conservation of Momentum
 Newton’s
3rd Law applies to
momentum as well as forces.
 Momentum will be conserved in
collisions.
–Conservation means the total
amount remains constant!
External Forces and
Internal Forces

An External Force is a force that is acting
outside your system. It may be acting on
your system or have no affect on your
system
 An Internal Force is a force that is acting
within your system between two objects
inside your system
How you define your system determines if
the forces are internal or external
Closed Isolated System


In order to observe the conservation of
momentum we need to have a Closed
Isolated System (CIS).
In a Closed Isolated System:
1. You must have a collection of objects
2. Nothing enters or leaves the system
3. There are no NET EXTERNAL FORCES
acting on your system
Conservation of Momentum
 If
no outside forces are acting on a
closed isolated system of objects,
then the total amount of momentum
of the system remains constant.
 pi = pf
 We observe the conservation of
momentum in Collisions!
Conservation of Momentum
in Collisions
Momentum can be conserved in:
 One Dimensional Collisions
 Two or Three Dimensional Collisions
 Elastic Collisions (1 or 2 dimensional)
 Inelastic Collisions (1 or 2 dimensional)
Elastic and Inelastic Collisions
To completely define elastic and
inelastic collisions requires energy
considerations (Chapter 11). Since we
have not gotten that far yet, we will say
for now;
 In an elastic collision the objects
separate after the collision
 In an inelastic collision the objects stick
together after the collision

Conservation of Momentum
Problems (1-D Collisions)
Count the number of objects in
your system before and after the
collision.
2. Substitute “mv” for every “p” in the
equation.
3. Put the numbers in and solve for
your unknown.
1.
Conservation of Momentum in a
1-Dimensional Elastic Collision

pi = pf
1. pAi + pBi = pAf + pBf
2. mAvAi + mBvBi = mAvAf + mBvBf
3. Substitute numbers in and solve!
Conservation of Momentum in a
1-Dimensional Inelastic Collision
 pi
= pf
1. pAi + pBi = pf
2. mAvAi + mBvBi = (mA + mB )vf
3. Substitute numbers in and solve!
Explosions
Momentum is conserved in Explosions
 Before the Explosion, the initial
momentum is zero (pi = 0)
 Therefore, the momentum after the
explosion must be zero (pf = 0)
 The momenta must be equal and
opposite after the explosion

Conservation of Momentum in
Two Dimensions
The conservation of momentum applies
to all closed isolated systems in one,
two and three dimensions
 Two Dimensional Conservation of
Momentum Problems can be done with
Vector Addition

Conservation of Momentum in
Two Dimensions
In doing two dimensional conservation of
momentum problems;
1. Draw your momentum vectors before
and after the collision
2. Treat pi = pf as a vector equation. Make
a vector diagram (triangle) and solve for
the momenta with vector addition
3. Solve for velocities after you find the
momenta
Conservation of Angular Momentum
Angular Momentum is also conserved
for Rotating Objects.
 Li = Lf
 Ii wi = If wf
 If the Rotational Inertia Changes then
the Angular Velocity must change to
keep the Angular Momentum (L)
constant.
