Download Lecture 21 - PhysicsGivesYouWings

Physics 101
April 11, 2017
Lecture 21
Momentum (Ch. 9)
Conservation of Momentum (Ch. 9)
Collisions (Ch. 9)
Elastic vs. Inelastic Collisions (Ch. 9)
Momentum
• Momentum, for a particle of mass
, is defined as:
and velocity
– Momentum is a vector.
– The SI unit of momentum is:
• For a system consisting of
momentum is:
particles, the
– If all particles have the same mass and velocity:
M. Afshar
2
Momentum Conservation
• Principle of momentum conservation: In the
absence of external interactions, the total
momentum of a system is constant in time.
– “Absence of external interactions”
means that the net external force is
zero:
.
– Conservation of momentum requires
that the net external force be zero;
internal forces are irrelevant.
M. Afshar
3
Momentum Conservation (cont.)
• Momentum conservation can be expressed
concisely as:
If
, then
.
• Other forms of momentum conservation:
If
, then
.
If
, then
.
• In discussing momentum, we say that a system is
“isolated” when
.
M. Afshar
4
How is Momentum Useful?
• Momentum is similar to energy.
– Momentum conservation relates initial momentum
to final momentum.
– Momentum conservation helps you calculate the
final velocity when the initial velocity is known.
• Four steps in using momentum conservation:
1. Choose your system carefully! It must be isolated.
2. Choose the initial time to be when all velocities are
known. This will help you calculate initial momenta.
3. Use
to solve for the final momentum.
4. Use
to solve for the final velocity.
M. Afshar
5
Practice Problem
A canon resting on frictionless ice fires a projectile parallel to
the surface. The canon recoils backward while the projectile
moves forward. The canon and the projectile have masses
and
. If the projectile has speed
, find the speed of the cannon immediately after it
has fired.
M. Afshar
6
Collisions
• An important class of problems in physics
involves collisions between two objects.
• We will model collisions as follows:
– Two objects are assumed to be isolated.
– Masses
and
are known and fixed.
– Initial velocities
and
are known.
– We wish to find final velocities
and
.
M. Afshar
7
Solving the Collision Problem
• There are two internal forces:
and
.
– These two forces are unknown and complicated.
• Since the internal forces are unknown, we cannot
calculate the acceleration of each object.
– So we cannot use kinematics.
• Since the internal forces are unknown, we cannot
calculate the work they perform.
– So we cannot use energy conservation.
• Momentum conservation does not depend on
internal forces.
– So we can use momentum conservation.
M. Afshar
8
Internal Forces
• Why are internal forces irrelevant?
– Short answer: Due to Newton’s third law, internal
forces cancel each other, and thus cannot change the
system’s momentum.
– Long answer:
, Since
, Since
, Since
, Since
M. Afshar
9
Practice Problem
Two isolated particles
and
collide
with initial speeds
and
. After
collision
rebounds with speed
.
a) What is the final velocity (speed and direction) of
?
b) Is the total kinetic energy of the system conserved?
a)
b) No.
(to the right)
,
.
M. Afshar
10
Elastic vs. Inelastic Collisions
• When two “isolated” bodies collide, momentum
is always conserved.
– Recall: If
, then
.
• Is kinetic energy conserved? Not always!
– Conserved for elastic collisions – kinetic energy
remains as kinetic energy.
• Example: Atomic collisions.
– Not conserved for inelastic collisions – kinetic energy
is transformed into other forms of energy.
• Example: Car collisions.
M. Afshar
11
Practice Problem
Two isolated particles
and
collide
with initial speeds
and
. The
collision is elastic. What is the velocity (speed and
direction) of each particle after the collision?
Cons. of :
Cons. of :
,
M. Afshar
12