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Momentum
Your Objectives
• You should understand impulse and momentum to relate mass,
velocity, and momentum for a moving object or to calculate the
total momentum for a system of bodies
• You should be able to relate impulse (J) to the change in linear
momentum and the average force acting on a body.
• You should be able to determine impulse and relate it to change
in momentum from Force-Time graphs
• You should be able to identify situations in which linear
momentum (or its components x&y) is conserved.
• You should be able to apply the conservation of momentum to
determine final velocity when two objects move along the same
line or at right angles, collide, and stick together, and to
calculate the change of kinetic energy of such an event.
Momentum
• A measure of how hard it is to
stop a moving object.
• Related to both mass and
velocity.
• Possessed by all moving
objects.
Calculating Momentum
• For one particle
p = mv
• For a system of multiple
particles
P = pi = mivi
• Momentum is a vector!
Which has the most
momentum?
A brief note about Kinetic Energy
• Kinetic energy is the energy of motion
Calculating Kinetic Energy (KE)
• Kinetic energy is very similar to momentum.
• Since an object must be in motion in order to
have kinetic energy, VELOCITY is a factor. (It
is THE MOST IMPORTANT factor)
• A large object in motion will have more
energy than a smaller object going the same
speed, so MASS will be a factor.
KE = ½ x mass x velocity2
KE = ½ mv2
KE is sometimes just referred to as K
Linear Momentum: Check for Understanding
Linear Momentum: Check for Understanding
http://www.bozemanscience.com/apphys-049-momentum
Impulse (J)
The product of an external
force and time, which results
in a change in momentum
•J = F t
•J = P
•Units: N s or kg m/s
3.
J
http://www.bozemanscience.com/ap
-phys-050-impulse
Impulse (J)
F(N)
3000
2000
area under curve
1000
0
0
1
2
3
4
t (ms)
2.
Answer: b
J
Law of Conservation of
Momentum
pb = pa
If the resultant external force on
a system is zero, then the vector
sum of the momenta of the
objects will remain constant.
pb = pa
• This means that
𝒎𝟏 𝒗𝟏𝒊 + 𝒎𝟐 𝒗𝟐𝒊 = 𝒎𝟏 𝒗𝟏𝒇 + 𝒎𝟐 𝒗𝟐𝒇
Collisions
• Collisions are governed by Newton's
laws.
• Newton’s Third Law tells us that the
force exerted by body A on body B in a
collision is equal and opposite to the
force exerted on body A by body B.
Collisions
During a collision, external forces are
ignored.
The time frame of the collision is very
short.
The forces are impulsive forces (high
force, short duration).
Collision Types
• Elastic (hard, no deformation)
– P is conserved, K is conserved
• Inelastic (soft; deformation)
– P is conserved, K is NOT conserved
• Perfectly Inelastic (stick together)
– P is conserved, K is NOT conserved
Elastic and Inelastic Collisions
Elastic and Inelastic Collisions
Elastic and Inelastic Collisions
6.4 Elastic and Inelastic Collisions
Golf and Momentum
Consider the
elastic collision
between the
club head and
the golf ball in
the sport of golf.
Golf and Momentum
Forces are on
the clubhead
and ball are
equal and
opposite.
Golf and Momentum
The acceleration
of the ball is
greater because
its mass is
smaller.
Pool and Momentum
Consider the
elastic collision
between a
moving ball and a
ball that is at rest
in the sport of
billiards.
Pool and Momentum
The balls
experience forces
which are equal
in magnitude and
opposite in
direction.
Pool and Momentum
Since the balls
have equal
masses, they
experience equal
accelerations.
Elastic and Inelastic Collisions: Check for Understanding
(Hint: See Special Case b)
Elastic and Inelastic Collisions: Check for Understanding
Elastic and Inelastic Collisions
Example : A rubber ball with a speed of 5.0 m/s collides head on
elastically with an identical ball at rest. Find the velocity of each
object after the collision.
Perfectly Inelastic Collision #1
An 80 kg roller skating grandma collides
inelastically with a 40 kg kid as shown.
What is their velocity after the collision?
Perfectly Inelastic Collisions #2
A train of mass
4m moving 5
km/hr couples
with a flatcar of
mass m at rest.
What is the
velocity of the
cars after they
couple?
Perfectly Inelastic Collisions #3
A fish moving at 2
m/s swallows a
stationary fish
which is 1/3 its
mass. What is the
velocity of the big
fish after dinner?
Explosion
• When an object separates suddenly, this is
the reverse of a perfectly inelastic
collision.
• Mathematically, it is handled just like an
ordinary inelastic collision.
• Momentum is conserved, kinetic energy is
not.
• Examples:
– Cannons, Guns, Explosions, Radioactive
decay.
An object splits apart (EXPLOSION)
• Examples:
– 2 objects together and moving push off from
each other
– Explosions
• Equations
– X direction:
• Pb1+2 = p1a + p2a
– Y direction
• 0 = p1a + p2a
– Or vice versa for the equations
Recoil Problem #1
A gun recoils when it is fired. The
recoil is the result of action-reaction
force pairs. As the gases from the
gunpowder explosion expand, the
gun pushes the bullet forwards and
the bullet pushes the gun backwards.
The Conservation of Linear Momentum: Check for Understanding
The Conservation of Linear Momentum: Check for Understanding
Explosion Example
• A 200 kg car is rolling at 10 m/s when
its fuel tank explodes into 4 pieces. If
40 kg of it goes forward at 25 m/s, 80
kg goes backwards, 30 kg goes to the
left at 10 m/s, and a piece goes to the
right. Find how much mass is going to
the right and how fast it is moving.
Also find out how fast the 80 kg piece is
moving backwards.
Elastic and Inelastic Collisions
Example : While standing on skates on a frozen pond, a student of
mass 70.0 kg catches a 2.00 kg ball travelling horizontally at 15.0
m/s toward him.
a) What is the speed of the student and the ball immediately after he
catches it?
b) How much kinetic energy is lost in the process?
Elastic and Inelastic Collisions: Check for Understanding
Elastic and Inelastic Collisions: Check for Understanding
Elastic and Inelastic Collisions: Check for Understanding
Elastic and Inelastic Collisions: Check for Understanding
6.4 Elastic and Inelastic Collisions: Check for Understanding
6.3 The Conservation of Linear Momentum
The Conservation of Linear Momentum
Momentum is conserved in an isolated system. The motion in
two dimensions may be analyzed in terms of the components
of momentum, which is also conserved.
If m1 and m2 are equal, they will split
apart at a 90 degree angle.
Inelastic collisions
Elastic and Inelastic Collisions
• In inelastic collisions, momentum is conserved but kinetic energy
is not.
• Collisions like the ones shown here, in which the objects stick
together , are called completely inelastic collisions.
• The maximum amount of kinetic energy lost is consistent with
the conservation of momentum.
Elastic and Inelastic Collisions: Check for Understanding
Elastic and Inelastic Collisions: Check for Understanding
Elastic and Inelastic Collisions
Sample Problem
Suppose three equally strong, equally
massive astronauts decide to play a game
as follows: The first astronaut throws the
second astronaut towards the third
astronaut and the game begins. Describe
the motion of the astronauts as the game
proceeds. Assume each toss results from
the same-sized "push." How long will the
game last?
Calculate momentum of the balls before
and after the collision.
2 m/s
3 m/s
2 kg
0 m/s
8 kg
Before
2 kg
50o
8 kg
V?
After