Momentum & Collisions

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Momentum and
Collisions
Physics Rapid Learning Series
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www.RapidLearningCenter.com/
© Rapid Learning Inc. All rights reserved.
Wayne Huang, Ph.D.
Keith Duda, M.Ed.
Peddi Prasad, Ph.D.
Gary Zhou, Ph.D.
Michelle Wedemeyer, Ph.D.
Sarah Hedges, Ph.D.
© Rapid Learning Inc. All rights reserved. - http://www.RapidLearningCenter.com 1

Learning Objectives
By completing this tutorial, you will:
„ Understand the
concepts of momentum
and impulse.
„ Mathematically
describe various types
of collisions.
„ Apply these ideas to 2-
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2
dimensional collisions.
Concept Map
Physics
Studies
Previous content
New content
Motion
Caused by
collisions Forces
F
Elastic
lli i
creates
Momentum
is a
Conserved
vector
Applied over time gives
Impulse
And
quantity
Vectors
Inelastic collisions
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Momentum
Although only moving objects possess
momentum its concept is very similar to
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momentum, inertia.
Momentum Definition
In a way, momentum is the motion equivalent of
inertia.
A large moving object is more difficult to stop
than a small moving object.
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Momentum describes or quantifies this
tendency to keep moving.

Momentum Formula
Momentum is the product of mass and velocity.
P = mv
momentum,
t
kg•m/s mass, kg velocity, m/s
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Substantial Momentum
A bullet may have a large momentum although
its mass is small.
P= mv
A tanker ship has a large momentum although its
velocity is small.
P
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P= mv

Changing Momentum
Obviously, all moving objects can be stopped
eventually .
Thus, their momenta are changed. This is usually
accomplished by changing the velocity of an
object.
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Changing Momentum with Force
To change the momentum of any object, you
must apply a force to it.
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This force must be applied over a period of time.
This could be a very short period of time, or a very
long period of time.

Impulse Formula
Impulse is equal to the product of force and time.
J = Ft
I l k / F N ti
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Impulse, kg•m/s Force, time, s
Impulse Observations
„ Impulse can also be defined as a change in
momentum, ΔP.
„ Notice that the units of momentum and impulse
are the same (kg•m/s).
„ This makes sense since impulse is just a change
in the amount of momentum.
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Impulse and Momentum Formula
Since impulse is defined as a change in momentum,
it can also be written as:
FΔt = Δ(mv)
Delta, change When an object’s
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in, time momentum changes, it
is usually the velocity
that changes, not the
mass.
No Bounce Interaction
A ball is falling towards the ground. It would take
some change in momentum, or impulse, to stop a
moving object.
P = mv
moving
Hits ground
Since v =0
P = 0
ΔP = mv
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g
ball not moving

Bouncing Interaction
However, it would take even more impulse to stop
the object, and then make it ricochet in the
opposite direction.
P = -mv
ball moving
Ball bouncing up
Hits ground
P = +mv
ΔP = 2mv
Twice as much
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down
as before
Impulse Example
During a rainstorm, drops come straight down
with a velocity of -15 m/s and hit a car roof. The
mass of rain hitting the car is .060 kg/s.
Assuming the rain comes to rest when it hits the
car, how much force is exerted on the car roof by
the rain?
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Solution
Begin from our impulse formula.
FΔt = Δ(mv) F = mΔ v
Δt
Change in
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F m(vf vi ) −
Δt
=
velocity, Δv
Calculation
F .060 kg ( 0 m/s 15m/s)
1sec
− −
=
F = .060 kg ( 15m/s)
1sec
two – signs will
F = .9 kgm F = .9N
s2
2
yield a +
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This is the force on the RAIN. The force on the car
must be in the opposite direction ( Newton’s 3rd
law). F= -0.9 N

Additional Question
If the raindrops were frozen with the same mass
and velocity, if they hit the car and ricocheted
upwards, how would that affect the force on the
car?
Because they would be
changing their momentum
more, due to the larger
change in their velocity, a
greater t i impulse l would ldb
be
required. Thus, more force
would be applied to the car.
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Conservation
of Momentum
In any type of collision or interaction, the
conservation of momentum will be a key
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idea to understanding the situation.

Conservation Laws
In all collisions or interactions, the momentum of
a system is always conserved.
You may have previously learned about
conservation of mass or energy from chemistry
class.
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Vector Quantity
Momentum is a vector quantity, direction must be
taken into account to see that momentum truly is
conserved.
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Rifle and Bullet Example
The rifle and bullet can be considered a system.
Before firing, they are both motionless and have a total
momentum of 0.
After firing, the total momentum still equals 0. The
rifle has momentum to the left, the bullet to the right.
The rifle has a much larger mass so its velocity is
less, but their momentum is still conserved.
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mv mv
Additional Observations
You may recall that Newton’s 3rd Law fits this
example too. The force on the rifle is equal
and opposite to the force on the bullet.
Reaction
Force
Action
Force
However, due to the difference in mass, the
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acceleration of the bullet is much greater than the
acceleration of the rifle. Acceleration is not a
conserved quantity.

Conservation Problems
When solving problems
involving the conservation of
momentum, the most
important thing to consider is:
Total momentum
b f lli i
Total momentum
ft lli i =
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before collision after collision
Explosion Sample Problem
A 300 kg cannon fires a 10 kg projectile at 200 m/s.
How fast does the cannon recoil backwards?
BOOM
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Solution Set up
The momentum of the projectile must be equal in
size to the momentum of the cannon.
They must be equal since they must cancel each
other out.
Pbefore = Pafter
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BOOM
Calculation
Pafter = Pbefore
mcannonvcannon + mprojvproj = 0
Before firing,
velocity = 0m/s.
cannon cannon proj proj
(300 kg) (vcannon) + (10kg) (200m/s) = 0
v 2000kgm/s cannon
300k
−
=
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300kg
vcannon = -6.67 m/s
Negative sign indicates
the cannon moves in
the opposite direction
to the projectile

Hit and Stick Sample Problem
Joe has a mass of 70 kg and is running at 7 m/s with
a football. He slams into 110 kg Biff who was
initially motionless. During this collision, Biff holds
onto and tackles Joe. This type of event may be
called a “hit and stick” collision. What is their
resulting velocity after the collision?
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Hit and Stick Solution
Pbefore = Pafter
PJoe PBiff PJoe+Biff + =
Biff’s initial velocity
is zero, so this term
drops out.
m1v1 +m2v2 = (m1 +m2 )v3
m1v1 = (m1 +m2 )v3
70kg(7m/s) = (70kg+110kg)v3
Since they stick
together, add their
masses.
Do math
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v3
70kg(7m/s) =
(70kg +
110kg)
v3 2.7m/s =
carefully
Since all velocities were in
the same direction, no –
signs are needed here.

Hit and Rebound Sample Problem
A 1 kg basketball rolls at +5 m/s and collides with
a stationary 4 kg bowling ball. The bowling ball is
given a velocity of +2 m/s. What is the velocity of
the basketball after the collision?
Find Vbasketball
after collision?
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Hit and Rebound Solution
Pbefore = Pafter
v=0m/s here
Pbasket +Pbowling = Pbasket +Pbowling
D Do math
th
carefully
m1v1 +m2v2 =m1v3 +m2v4
m1v1 =m1v3 +m2v4
(1kg)5m/s = (1kg)v + (4kg)2m/s
- sign shows
the basketball
is moving in
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(1kg)5m/s - (4kg)2m/s = (1kg)v
the opposite
direction
(1kg)5m/s - (4kg)2m/s = v
= −3m/s
(1kg)

Summary of Collisions
Explosion: one object breaking into more objects.
0 =mv +mv + ...
Hit and stick: one object striking and joining to
the other.
+ = mv1 mv2 (m1 +m2 )v3
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Hit and rebound: one object striking and bouncing
off of the other.
+ = m1v1 m2v2 m1v3 +m2v4
Subscripts
In the momentum conservation formulas, various
subscripts are often used to keep track of the
various objects. Don’t be confused by them, they
are used to help keep track of the variables.
Some examples:
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m1v1=1st mass and 1st velocity of that mass
mAv’=1st mass and the 2nd velocity of that mass
m2xv2x=2nd mass and 2nd velocity only in the x direction

Elastic and Inelastic
Collisions
All collisions or interactions can be
described as elastic or inelastic
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collisions.
Elastic Collision
„ Momentum is conserved.
„ The objects colliding aren’t deformed or
smashed
„ Thus no kinetic energy is lost; kinetic energy is
conserved also.
„ Ex: billiard ball collisions
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Inelastic Collision
„ Momentum is conserved.
„ Kinetic energy is lost.
„ The energy may be transformed into sound
sound,
deformation of materials, flying debris, etc.
„ Often objects interlock or stick together.
„ Objects are also often deformed or crunched.
„ Example: Car crash
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Joe and Biff’s Collision
Is our previous football tackle an example of an
elastic or inelastic collision?
Before Collision After Collision
m1 = 70 kg
v1 = 7 m/s
m2 = 110 kg
v2 = 0 m//
s
m1+2 = 180 kg
v3 = 2.7 m/s
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We know momentum is conserved, but if kinetic
energy is conserved too, then it would be an elastic
collision.

Calculation of Kinetic Energy
KE mv
2
2
=
KE formula from a
previous tutorial
Before collision: After collision:
KE 180kg(2.7m/s)
2
2
KE 70kg(7m/s)
2
2 =
=
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KE = 1715J KE = 656J
Over 1000 J of energy are lost due to friction, heat,
deformation, etc! Definitely an inelastic collision.
Additional Directions
Many collisions involve motion in more
than one direction. The same concepts
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can be applied to describe these new
situations.

2-Dimensional Collisions
Collisions do not always take place in a nice neat
line. Often, collisions take place in 2 or 3
dimensions:
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Although the mathematics needed to show this
may be somewhat lengthy, the general idea can
easily be conveyed.
2-Dimensional Movement
One ball collides into another. By using momentum
vector components, you can predict the result:
Before impact:
After impact:
BAM
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Total P before
Y components cancel out
X components add up to previous P

Consider the Components
It’s easiest to break the momentum into X and
y components. Since momentum is always
conserved:
PX before =PX after
P P
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PY before = PY after
Sample 2-Dimensional Problem
Two pool balls, each 0.50 kg collide. Initially, the
first moves at 7 m/s, and the second is
motionless. After the collision, the first moves 40o
to the left of its original direction, the second
moves 50o to the right of its original direction.
Find the velocitiy of B after the collision.
A B
A After Collision
40o
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Before Collision
B
50o

Consider the Components
The X and Y components of momentum are both
conserved. You can visualize this several ways:
A
A B
B
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After the collision, the sum of the X components
equals the original momentum.
The y components cancel out since there was no
momentum in that direction originally.
Add the Vectors
Without using components, it can also be noticed
that both momentum vectors after the collision add
up to the original momentum vector:
A B
A
Resultant equal to Pbefore
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B
Remember that vectors can be moved anywhere
as long as their magnitude and relative direction
are unchanged. They are added tip to tail.

Problem Solution
sinθ = opp
hyp
sin 40o = PB
P
Use trig to find the
momentum of ball B. Then
find its velocity.
A B
40o
3.5 kgm/s
PB 2.25 kgm/s =
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Pbefore =
mv
= =
.5kg(7m/s) 3.5kgm/s
P =
m v
B B B
2.25 kgm/s =
.5 kg v
v 4.5 m/s
B
B
=
Only the Vectors are Conserved
When observing the
conservation of
momentum, be sure to
remember that momentum
is a vector quantity.
Adding the values as
scalars won’t work. The
direction must be
accounted for by vector
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diagrams, trigonometry,
etc.

Learning Summary
Collisions
b l ti
Momentum =
mass x
All momentum
concepts can
also be
may be elastic
or inelastic
velocity
applied to
more than 1
dimension
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In any collision,
momentum is
conserved
Impulse = Force
x time
Congratulations
You have successfully completed
the core tutorial
Momentum and
Collisions

Chemistry :: Biology :: Physics :: Math
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