Chapter 9: Linear Momentum and Collisions

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Describe momentum

-the product of mass times velocity
-"velocity enhanced"

What is the equation that describes momentum?

p = mv

What are the units of momentum?

-kg * m/s

What are the units of force? Long and short version please!

-kg * m/s^2
-newtons

What are the units of work and energy? Long and short version please!

-kg * m^2/s^2

What does momentum allow for?

-the fundamental expression of Newton's second law

What is the relationship between Fnet, Δp, and Δt?

Fnet = Δp/Δt

Describe this relationship in words

Fnet tells you the rate of change of the momentum over time

Therefore, if you are applying a net force, what are you changing?

-the momentum over time

Why is Fnet = ma a special case of Fnet = Δp/Δt?

-because it assumes that mass is constant

If we assume that mass is constant, what is Δp/Δt equal to?

Fnet = Δp/Δt = Δmv/Δt = mΔv/Δt = ma

What if mass is not constant?

-examination requires calculus so we won't bother with it!

What is impulse equal to?

I = (Fav)(Δt)

What does it describe?

-a change in momentum

Is momentum conservative or nonconservative?

-conservative

What is the definition of linear momentum?

-the product of the mass m and the velocity of an object

What specifically is constant LINEAR momentum?

-the object of mass m that is MOVING IN A STRAIGHT LINE with a velocity v

Is velocity a vector?

-yes

Therefore what does it have?

-both a magnitude and a direction

If velocity is a vector, what is momentum?

-a vector

In order to increase the momentum from a negative value to zero, what is it necessary to add?

-a positive momentum

What is Newton's original statement of the second law?

Fnet = Δp/Δt

Describe this law

-the net force is the rate of change of momentum with time

In what case does Fnet = ma?

-when mass is constant

Is the force required to hold an umbrella during hail the same as, more than, or less than the force required in the rain?

-the force required is greater in the hail

Why?

-when hailstones hit an umbrella, they bounce back upward
-as a result, change in momentum is greater hail
-therefor, the impulse and the force are greater with hail

Is linear momentum a conserved quantity?

-yes

If the net force acting on an object is zero, is its momentum conserved or not conserved?

-conserved

What do internal and external forces occur in?

-action-reaction pairs

What must internal force always sum to?

-zero

Why?

-because the forces in an action-reaction pair are equal and opposite

Therefore, what is the sum of the net force acting on a system?

-the sum of the external forces acting on it

To summarize, what effect do internal forces have on the net momentum of a system?

-no effect

If the net external force acting on a system is zero, what about the momentum?

-the net momentum is conserved

What do these statements apply to?

-the net momentum of a system

What is a collision?

-a situation in which two objects strike one another in which the net external force is zero or negligibly small

What happens to the momentum of a system during a collision?

-the momentum of a system is conserved

What about kinetic energy?

-most or even all of a system's kinetic energy may be converted to other forms during a collision

What are the two types of collisions?

-elastic collisions
-inelastic collisions

When are conditions said to be elastic?

-when KEf = KEi
-->KE is conserved

When are conditions said to be inelastic?

-when KEf is not equal to KEi
-->KE is not conserved

What typically occurs to KE?

-it decreases

Due to losses associated with...?

-sound, heat and deformation

Provide an example of when KE may increase as a result of a collision?

-if the collision sets off an explosion

To summarize, describe conditions of inelastic conditions in terms of momentum and kinetic energy

-momentum of a system is conserved
-kinetic energy of a system is not conserved

What is a completely inelastic collision?

-a collisions where objects stick together afterwards

In a completely inelastic collision, what occurs to the KE?

-the maximum amount of kinetic energy is lost

What is the relationship between initial velocity and final velocity in a system that has just experienced an inelastic collision?

final velocity = 1/2 initial velocity

Using momentum to evaluate a collision between two objects allows us to bypass what?

-the details of the collision itself
-->doesn't matter how long the objects spent in contact
-->doesn't matter how much force objects exerted on each other

If you want to change the momentum of two objects involved in a collision, how must you do so?

-using external forces

Why?

-because the momentum of a system that experiences no external forces is conserved

For what type of conditions is momentum most useful for?

-for multi-object system with unknown internal forces

What must Fnet external be for us to use momentum?

-zero

If two people are standing on ice, what is their momentum?

-zero

If the two people standing on ice push off of each other, what is the relationship between their momenta?

-are equal in magnitude and opposite in direction

When is the momentum of a system conserved?

-when the net external force on a system is equal to zero
-->the only forces that are present are internal forces

When is mechanical energy conserved?

-when work non-conservative is equal to zero
-->only work is conservative

What are the two types of mechanical energy that we work with the most?

-KE and PE

What are the three types of collisions?

-elastic
-partially inelastic
-completely inelastic

Is momentum conserved during elastic collisions?

-yes

Are forces internal, external, or both during elastic collisions?

-internal

Are forces conservative or nonconservative during elastic collisions?

-conservative

If forces are conservative, what is the relationship between KE initial and KE final?

KE initial = KE final

KE is therefore...?

-conserved

Before collision, will the energy of the system be predominantly KE or PE?

-KE

During the collision?

-PE
-->energy is absorbed during the collision

After the collision?

-KE
-->energy that was absorbed during the collision is returned to the system

What is the potential energy of gravity equal to during elastic collisions? Why?

-the potential energy of gravity is equal to zero because collisions typically occur on a flat surface

Are forces internal, external, or both during inelastic collisions?

-internal

Is momentum conserved during inelastic collisions?

-yes

Is KE conserved?

-no
--> KE system final < KE system initial

Where does the energy go?

-energy is lost to the environment

What is the difference between partially inelastic conditions and completely inelastic collisions?

-the degree to which KE is lost
-->in both instances, KE system final < KE system initial

What event of importance occurs during completely inelastic conditions?

-so much energy is lost that objects will not separate after the collision

What energy forms does mechanical energy include?

-KE and PE

During an elastic collision, is momentum conserved?

-yes

Why?

-there are no external forces acting on objects within the system

During an elastic collision, is KE conserved?

-yes

Why?

-no non-conservative work takes place

Therefore, how would you describe the forces that act during an elastic collision?

-internal and conservative

Is momentum conserved during an inelastic collision?

-yes

Why?

-there are no external forces acting on objects within the system

Is KE conserved during an inelastic collision?

-no

Why not?

-there are non-conservative forces at work

Therefore, how would you describe the forces that act during an inelastic collision?

-internal and non-conservative

Describe the two 2 subclasses of inelastic collisions

1. partially inelastic collisions (objects bounce off of each other, minimal KE loss)
2. completely inelastic collisions (objects stick together, maximal KE loss)

If two people are at rest on ice (horizontal frictionless surface), then push off of each other...is momentum conserved?

-yes

Why?

-no external forces act on the two people

Is KE conserved?

-no

Why not?

-when two people are standing, KE = 0
-after the collision, KE > 0

Therefore, what is the relationship between KE final and KE initial?

KE final > KE initial

What must have taken place?

-positive non-conservative work

Where did this work come from?

-the chemical energy from the persons

What then was added to the system?

-mechanical energy in the form of KE

Therefore, how would you describe the forces that act during this collision?

-internal and non-conservative

If a ball is free falling under the influence of gravity, is momentum conserved?

-no it is not

Why not?

-the force of gravity is external

Is mechanical energy conserved?

-yes

Why?

-the potential energy of gravity is converted to KE as the ball drops
-->both are mechanical forces so E is conserved

What will be the effect of momentum conservation if Earth is added to the system?

-momentum will be conserved

An object slides along a table and hits a friction patch. Is momentum conserved?

-no

Why not?

-the friction patch is an external force acting on the object

Is KE conserved?

-no

Why not?

-force of kinetic friction is a non-conservative force

What is the center of mass of an object?

-the only part of an extended object that obeys point particle physics

What is an extended object?

-an object that has size and shape

What type of motion does the COM exhibit?

-linear/translational motion

What type of motion does the rest of the object exhibit?

-rotational motion

What are the four ways to determine the COM?

1. Symmetry Method
2. Balance Method
3. Hanging Method
4. Calculation Method (1D and 2D)

What is the midpoint of an object referred to as?

-geometric center

Is the geometric center the COM for asymmetrical objects?

-no, the COM is not likely to be the midpoint/geometric center of an object that is asymmetric

When objects sit in a 2D plane, what is the first step to calculating the COM?

-figure out how much mass is at each coordinate

Then?

-pick one object to set as the origin
-->it is easier to plug in zero to an equation than non-zero values

Then?

-determine the x and y coordinates of other masses and any connecting rods (or similar objects)

Then?

-take the x values, find the x COM

Then?

-take the y values, find the y COM

What type of collisions are those in which both momentum and kinetic energy are conserved?

-elastic collisions

Two objects (A & B), initially sliding to the right and left respectively on a horizontal frictionless table, both stop after a head on collision. During this collision, the momentum of the A + B system is...?

-conserved

During this collision, the ME of the A + B system is...?

-decreased

If object is is more massive than object B, this means the initial momentum of the system is....?

-zero

Bob the snowboarder speeds up while descending a hill with friction. He then flies off a cliff (we neglect air resistance). While BOB is descending the initial hill, what happens to his KE?

-increases

Why?

-it is given that Bob speeds up
-KE depends on speed, so it will decrease

What will happen to his mechanical energy?

-since friction acts, ME is removed to heat

What will happen to the size of his momentum?

-increases (since it is dependent on mv and Bob is speeding up)

While Bob is in the air, what will happen to his KE?

-increases

Why?

-because Bob continues to speed up as he is in free fall (due to gravity)

While Bob is in the air, what will happen to his ME?

-stays the same

Why?

-because friction is no longer at play

If we were to compare an equal height drop while Bob was on the hill vs. through the air, in which case would the increase in speed be greater?

-while Bob was in the air

Why?

-while the PE loss would be the same in both cases, it would ALL go into KE while in the air (since ME is conserved) while only SOME of it would go into KE while on the hill (some PE went to heat due to the friction force, not to KE)

If you drop your keys, their momentum increases as they fall. Why is the momentum of the keys not conserved? Does this mean that the momentum of the universe increases as the keys fall? Explain

-the momentum of the keys increases as they fall because a net force acts on them
-the momentum of the universe is unchanged because an equal and opposite force acts on the Earth

A system of particles is known to have zero momentum. Does it follow that the KE of the system is also zero? Explain.

-if the kinetic energy is zero the speeds must be zero as well (due to the fact that mass itself cannot be zero). This means that the momentum is also zero.

Crash statistics show that is safer to be riding in a heavy car in an accident than in a light car. Explain in terms of physical properties?

-when a heavy object and a light object collide they exert equal and opposite forces on one another
-since the light object has more mass, its acceleration is greater. this can result in more severe injuries for the light vehicle

An object at rest on a frictionless surface is struck by a second object. Is it possible for both objects to be at rest after the collision? Explain.

-no
-the fact that the initial momentum of the system is nonzero means that the final momentum also must be nonzero
-thus, it is not possible for both objects to be at rest after the collision

In what scenario can two objects on a horizontal frictionless surface have a collision in which all the initial KE of the system is lost?

-when the objects have momenta of equal magnitude prior to the collision
-->if these objects collide in a head-on, completely inelastic collision, they will be at rest after the collision
-->in this case, all of the initial kinetic energy is converted into other forms of energy

Can two such objects have a collision in which all the initial momentum of the system is lost?

-no
-in order for its momentum to change, an external force must act on teh system

At the instant a bullet is fired from a gun, the bullet and the gun have equal and opposite momenta. Which object--the bullet or the gun--has the greater KE? Explain.

-the KE of the bullet is much greater than the gun
-if the momenta of the gun and the bullet are equal immediately after firing, we know that the bullet must have a greater v since its mass is much smaller than that of the gun
-accordingly, if we calculate KE from the velocities of the objects, the KE for the bullet will be greater (since the v term is squared)

A block of wood is struck by a bullet. Is the block more likely to be knocked over if the bullet is metal and embeds itself in the wood, or if the bullet is rubber and bounces off the wood?

-the rubber bullet is more likely to knock the block over
-the reason is that the change in momentum is twice as great when an object rebounds was it is when the object is simply brought to rest

A net force of 200 N acts on a 100 kg boulder, and a force of the same magnitude acts on a 100 g pebble. Is the change of the boulder's momentum in one second greater than, less than, or equal to the change of the pebble's momentum in the same time period?

-the boulder and the pebble have the same rate of momentum change, since the same force acts on both objects
-(force is the rate of change of momentum)

A friend tosses a ball fo mass m to you with a speed v. When you catch the ball, you feel a noticeable sting in your hand. If you now catch a ball of mass 2m and speed v/2, is the sting you feel greater than, less than, or the same as that felt when you caught the first ball.

-the two balls have the same momentum. however, the first ball has a KE equal to 1/2mv^2 and the second ball has half that much KE
-->thus, less energy is dissipated in stopping the second ball, so it has less "sting"

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