# Chapter 9: Linear Momentum and Collisions

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### Describe momentum

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

p = mv

-kg * m/s

-kg * m/s^2
-newtons

-kg * m^2/s^2

### What does momentum allow for?

-the fundamental expression of Newton's second law

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!

I = (Fav)(Δt)

### What does it describe?

-a change in momentum

-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

-yes

### Therefore what does it have?

-both a magnitude and a direction

-a vector

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

-a positive momentum

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

-yes

-conserved

### What do internal and external forces occur in?

-action-reaction pairs

-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

-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

-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

-zero

-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

-KE and PE

### What are the three types of collisions?

-elastic
-partially inelastic
-completely inelastic

-yes

-internal

-conservative

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

KE initial = KE final

-conserved

-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

-internal

-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

-KE and PE

-yes

### Why?

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

-yes

### Why?

-no non-conservative work takes place

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

-internal and conservative

-yes

### Why?

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

-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)

-yes

### Why?

-no external forces act on the two people

-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

-no it is not

### Why not?

-the force of gravity is external

-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

-no

### Why not?

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

-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

-conserved

-decreased

-zero

-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)

-increases

### Why?

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

-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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