Physics - Chapter 6 - Momentum and Collisions
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Transcript of Physics - Chapter 6 - Momentum and Collisions
Lesson 6-1Momentum and Impulse
Linear Momentum
Think of a batter hitting a baseball When the batter swings and makes contact, the
ball changes velocity very quickly We could use kinematics to study the motion of the ball We could use Newton’s Laws to explain why the ball
changes direction
We are now concerned with the force and duration of the collision
Momentum
Momentum describes an object’s motion To describe force and duration of a collision, we
must first start with a new concept
Momentum This word is used in everyday conversation, and
means about the same thing in physics
Momentum
We might say a semi-truck has a large amount of momentum
Compared to the semi-truck, a person would have a small amount of momentum
Linear momentum directly relates an object’s velocity to the object’s mass
Momentum (P) P=mv
Momentum
Momentum is a vector quantity, with the vector matching the direction of the velocity
The SI unit is kg∙m/s
Bowling
If you bowl with a light ball, you have to throw the ball pretty fast to make the pins react
A heavier ball will allow a good pin reaction with a lower velocity Because of the added mass
Example 209 Practice 209
Change in Momentum
Recall: change in velocity takes an acceleration and time
If there is an acceleration, there exists a net force
Since P depends on velocity, ΔP requires Force Time
Change in Momentum
Say there is a ball rolling on the ground You must use a large force to stop a fast rolling
ball You could use a smaller force to stop a slower
rolling ball
Imagine catching a basketball A faster pass stings the hands a bit A softer pass causes almost no feeling
Newton’s Second Law
Imagine a toy fire truck and a real fire truck sitting at the top of a hill If they both begin to roll down the hill, which will have the
greater velocity? Recall: all objects fall due to gravity at the same rate
But which would require the greater force to stop Examples like this show us that P is closely related
to force
Newton’s Second Law
When Newton first wrote his second law (F=ma), he wrote it as
Fp
t= ∆
∆
Fp
t= ∆
∆F
mv
tmv
tma= = =
Impulse – Momentum Theorem
This states a net external force, F, applied for a certain time interval, Δt, will cause a change in the object’s momentum equal to the product of the force and time interval
In simpler terms, a large constant force will cause a rapid change in P
A small constant force would take a much longer time to cause a change in P
F t p∆ ∆= F t p mv mvf i∆ ∆= = −
Impulse – Momentum Theorem
The Impulse – Momentum theorem explains why “follow through” is so important in many sports such as baseball, basketball, and boxing
When a baseball player hits a baseball and “follows through” the ball is in contact much longer and the force is applied over a greater period of time
If the player does some sort of check swing, the force is applied over a smaller period of time
Sample 211 Practice 211 Sample 212 Practice 213
Impulse – Momentum Theorem
Change in momentum over a longer time requires less force Engineers use the impulse – momentum theorem
to design safety equipment Safety gear aims to reduce the force exerted on
the body during a collision
Impulse – Momentum Theorem
Think of jumping on a trampoline Do you think you could jump that high and land on
the ground and not get hurt? The impact with the ground is sudden and occurs
over a short period of time The impact with the trampoline is the same, but
occurs over a longer period of time Longer time interval = less force
Lesson 6-2Conservation of Momentum
Billiards
In a game of pool: The object ball is stationary The cue ball is moving
During the collision, the object ball gains momentum and the cue ball loses the same amount of momentum
The momentum of each ball changes during the collision but total momentum remains constant
Conservation of Momentum
Since the momentum of the two billiard balls remains constant after the collision we say momentum is conserved
P P P PAi Bi Af Bf+ = +
m v m v m v m vi i f f1 1 2 2 1 1 2 2+ = +
Conservation of Momentum
As we just discussed, momentum is conserved during collisions
Momentum is also conserved when objects push away from each other
Conservation of Momentum
Imagine you stand on the ground and jump up It seems as if momentum is not conserved because you
leave the ground with a velocity Recall, the Earth does move away when you jump (a very
small distance), so total momentum is conserved in reality You exert a downward force on the Earth and the Earth
exerts an upward force on you Total momentum is ZERO
Conservation of Momentum
The reason total momentum is zero when two objects push apart is based on sign
The objects have the same amount of momentum
But in opposite directions So when the two momentums are summed, the
result is zero
Sample 218 Practice 219
Relation to Newton’s Third Law
Consider two bumper cars of m1 and m2
describes the change in momentum is one of the cars
and
F1 is the force that m1 exerts on m2
F2 is the force that m2 exerts on m1
F t p∆ ∆=
F t m v i1 1 1∆ = F t m v i2 2 2∆ =
Relation to Newton’s Third Law
Since the only forces are from the two bumper cars, Newton’s third law tells us the forces must be equal and opposite
Additionally, the impulse (time of collision) is equal and opposite for both cars
This means EVERY interaction between the two cars is equal and opposite and can be expressed by:
m v m v m v m vi f i f1 1 1 1 2 2 2 2− = − −d i
Relation to Newton’s Third Law
The equation says ‘if the momentum of one object decreases during a collision, the momentum of another object will increase by the same amount’
At all times during a collision the forces are equal and opposite The magnitudes and directions are constantly changing The value we use for force is equal to average force
Lesson 6-3Elastic and Inelastic Collisions
Everyday Collisions
You see collisions everyday In some collisions, the objects stick together and travel as
one mass In another type of collision, the objects hit and bounce
apart
In either case, total momentum is conserved KE is usually not conserved because some energy
is lost to heat and sound energies
Perfectly Inelastic Collisions
When two objects collide and move together as one mass, the collision is called perfectly inelastic A good example of this type of collision is a
meteor hitting the Earth Perfectly inelastic collisions are easy to
analyze in terms of momentum because the two objects essentially become one after the collision
Perfectly Inelastic Collisions
The final mass is equal to the combined mass of the two objects
The two objects travel together with one final velocity after the collision
Studied with the following equation:
m v m v m m vi i f1 1 2 2 1 2+ = +( )
Perfectly Inelastic Collisions
KE does not remain constant in an inelastic collision
KE is lost due to sound, internal energy, and heat of fusion
Elastic vs Inelastic
The phenomena of fusion helps us to understand the difference between elastic and inelastic collisions
When we think of something that is elastic (a rubber band, a bungee cord, a spring) we think of something that returns to its original shape
During an elastic collision, the objects maintain their original shapes
Elastic vs Inelastic
Objects in inelastic collisions do not maintain their original shapes as they form a new mass after the collision
We can calculate the loss of KE with the conservation of KE formula KEnet = KEf – Kei
Sample 225 Practice 226
Elastic Collisions
When a soccer player kicks a soccer ball, the ball and the player’s foot remain separate
Since there are no shape changes or deformities, the is no change in KE
As with any collision, total momentum is conserved
In the Real World
It should be mentioned that there is no such thing as a perfectly inelastic or perfectly elastic collision in the real world
Objects do not hit into each other and fuse together and move as one object
Objects do not bounce off of each other without loss of KE KE lost to heat, sound, deformation
In the Real World
So that means that most collisions fall into a third category called inelastic collisions (note: not perfectly inelastic) This is where objects collide, make noise, give off heat, do
not stick together, and travel in another direction with separate velocities
These are impossible to study to complete exactness To study these types of collisions, we simplify things
Elastic Collisions
KE is conserved in elastic collisions There are instances that are very, very close to
perfectly elastic collisions Bowling ball into bowling pins Golf club hitting a golf ball
In these instances, we assume total KE and total momentum remain constant throughout the collision
Elastic Collisions
We can study elastic collisions with the following formulas:
Sample 228 Practice 229
m v m v m v m vi i f f1 1 2 2 1 1 2 2+ = +
1
2
1
2
1
2
1
21 12
2 22
1 12
2 22m v m v m v m vi i f f+ = +