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∆ ∆= = −


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


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


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+ = +


As we just discussed, momentum is conserved during collisions

Momentum is also conserved when objects push away from each other


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


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∆ =


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


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


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+ = +( )


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


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:


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+ = +

physics - chapter 6 - momentum and collisions

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