- [Instructor] What does momentum mean?
The definition of momentum is the mass times the velocity.
So the formula is simple, it's just m times v.
And why do we care about momentum?
We care about momentum because if there's no net
external force on a system,
the momentum of that system will be conserved.
In other words, the total initial momentum of that system
would equal the total final momentum of that system.
So momentum will be conserved
if there's no net external force.
And momentum is a vector, that means it has components.
The total momentum will point in the direction
of the total velocity, and the momentum in each direction
can be conserved independently.
In other words, if there's no net force in the y direction,
then the momentum in the y direction will be conserved,
and if there's no net force in the x direction,
the momentum in the x direction will be conserved.
Since the momentum is m times v, the units are
kilograms times meters per second.
So what's an example problem involving momentum look like?
Let's say two blocks of mass 3M and M
head toward each other, sliding over a frictionless surface
with speeds 2V and 5V, respectively,
and after the collision they stick together.
Which direction will the two masses head
after the collision?
To figure this out, we can just ask what direction is the
total momentum of the system initially.
Since momentum's gonna be conserved, that'll have to be the
direction of the momentum finally.
So the momentum of the 3M mass
is going to be the mass, which is 3M,
times the velocity, which is 2v,
so we get a momentum of 6Mv.
And the momentum of the mass M is gonna be the mass M,
times the velocity, which is negative 5v.
Momentum is a vector, so you can't
forget the negative signs.
Which gives a momentum of negative 5Mv.
So the total initial momentum of the system would be
6Mv plus -5Mv, which is one Mv, and that's positive,
which means the total momentum, initially, is to the right.
That means, after the collision,
the total momentum will also have to be to the right,
and the only way that could be the case
if these two masses joined together, is for the total
combined mass to also move to the right.
What does impulse mean?
The impulse is the amount of force exerted on an object
or system multiplied by the time during which
that force was acting.
So in equation form, that means J, the impulse, is equal to
the force multiplied by how long that force was acting.
And the net impulse is gonna be equal to the net force times
the time during which that net force was acting.
And this is also going to be equal to the change
in momentum of that system or object.
In other words, if a mass had some initial momentum
and ends with some final momentum,
the change in momentum of that mass,
p final minus p initial, is gonna equal the net impulse,
and that net impulse is gonna equal the net force
on that object multiplied by
the time during which that force was acting.
And since impulse is a change in momentum,
and momentum is a vector, that means impulse is also
a vector, so it can be positive and negative.
And the units are the same as momentum,
which is kilograms times meters per second.
Or, since it's also force times time, you could write
the units as Newtons times seconds.
So what's an example problem involving impulse look like?
Let's say a bouncy ball of mass M is initially
moving to the right with a speed 2v.
And it recoils off a wall with a speed v.
We want to know, what's the magnitude of the impulse
on the ball from the wall?
So the impulse, J, is going to be equal
to the change in momentum.
The change in momentum is p final minus p initial,
so the final momentum is gonna be the mass times the final
velocity, but this velocity's heading leftwards,
so you can't forget the negative sign,
minus the initial momentum, which would be M times 2v,
which gives a net impulse of -3Mv.
This makes sense.
The net impulse has to point in the
same direction as the net force.
This wall exerted a force to the left,
that means the impulse also points left,
and has a magnitude of 3Mv.
If you get a force versus time graph,
the first thing you should think about is that
the area under that graph is going to equal
the impulse on the object.
So if you graph the force on some object as a function of
time, the area under that curve is equal to the impulse.
Just be careful, since area above this time axis
is going to count as positive impulse, and area underneath
the time axis would count as negative impulse,
since those forces would be negative.
Why do we care that the area's equal to the impulse?
Well, if we can find the area, that would equal the impulse,
and if that's the net impulse on an object,
that would also equal the change in momentum of that object.
Which means we could figure out
the change in velocity of an object.
So what's an example problem involving impulse
as the area under a graph look like?
Let's say a toy rocket of mass two kilograms
was initially heading to the right
with a speed of 10 meters per second,
and a force in the horizontal direction
is exerted on the rocket, as shown in this graph,
and we want to know, what's the velocity of the rocket
at the time t equals 10 seconds?
To figure that out, we'll figure out
the area under the curve.
This triangle would count as positive area.
This triangle would count as negative area.
And since this triangle is just as positive
as this triangle is negative, these areas cancel completely.
And the only area we'd have to worry about is the area
between eight seconds and 10 seconds.
This is going to end up being a negative area,
since the height of the rectangle is -30,
and the width of the rectangle is going to be two seconds.
This gives an impulse of -60 Newton seconds.
So if the impulse on this object is -60 Newton seconds,
that's going to equal the change in momentum of that object.
How much momentum did this object start with?
The initial momentum of this object
is going to be two kilograms times the initial velocity,
which was 10 meters per second to the right,
which is positive 20 kilogram meters per second.
So if the initial momentum of the rocket is positive 20
and there was a change in momentum of -60,
the final momentum just has to be -40.
Or in other words, since the change in momentum would have
to be the final momentum minus the initial momentum,
which was positive 20, we could find the final momentum by
adding 20 to both sides, which would give us -60
plus 20, which is -40.
What's the difference between an elastic
and an inelastic collision?
What we mean by an elastic collision is that
the total kinetic energy of that system
is conserved during the collision.
In other words, if a sphere and a cube collide,
for that collision to be elastic,
the total kinetic energy of the sphere plus
the kinetic energy of the cube before the collision
would have to equal the kinetic energy of the sphere
plus the kinetic energy of the cube after the collision.
If the total kinetic energy before the collision
is equal to total kinetic energy after the collision,
then that collision is elastic.
It's not enough for the system to just bounce of each other.
If two objects bounce, the total kinetic energy
might not be conserved.
Only when the total kinetic energy is conserved
can you say the collision is elastic.
For an inelastic collision, the kinetic energy is not
conserved during the collision.
In other words, the total initial kinetic energy
of the sphere plus cube would not equal
the total final kinetic energy of the sphere plus cube.
Where does this kinetic energy go?
Typically, in an inelastic collision,
some of that kinetic energy is transformed
into thermal energy during the collision.
While masses could bounce during an inelastic collision,
if they stick together, the collision is typically called
a perfectly inelastic collision,
since in this collision you'll transform the most
kinetic energy into thermal energy.
And when two objects stick together, it's a surefire sign
that that collision is definitely inelastic.
So what's an example problem that involves elastic
and inelastic collisions look like?
Let's say two blocks of mass 2M and M head toward each other
with speeds 4v and 6v, respectively.
After they collide, the 2M mass is at rest,
and the mass M has a velocity of 2v to the right.
And we want to know,
was this collision elastic or inelastic?
Now you might want to say that,
since these objects bounced off of each other,
the collision has to be elastic, but that's not true.
If the collision is elastic, then the objects must bounce,
but just because the objects bounce
does not mean the collision is elastic.
In other words, bouncing is a necessary condition
for the collision to be elastic, but it isn't sufficient.
If you really want to know whether a collision was elastic,
you have to determine whether the total kinetic energy
was conserved or not.
And we can figure that out for this collision
without even calculating anything.
Since the speed of the 2M mass decreased,
the kinetic energy of the 2M mass went down.
And since the speed of the M mass also decreased
after the collision,
the kinetic energy of the mass M went down, as well.
So if the kinetic energy of both masses go down,
then the final kinetic energy after the collision
has to be less than the initial kinetic energy.
Which means kinetic energy was not conserved,
and this collision had to be inelastic.
One final note, even though kinetic energy
wasn't conserved during this process,
the momentum was conserved.
The momentum will be conserved
for both elastic and inelastic collisions.
It's just kinetic energy that's not conserved
for an inelastic collision.
How do you deal with collisions in two dimensions?
Well the momentum will be conserved for each direction
in which there's no net impulse.
If there's no net impulse in both directions,
then the momentum in both directions
will be conserved independently.
In other words, if there's no net force in the x direction,
the total x momentum has to be constant,
and if there's no net force in the y direction,
the total momentum in the y direction has to be constant.
So in other words, if two spheres collide
in a glancing collision, the total momentum
in the x direction initially should equal
the total momentum in the x direction finally,
if there's no net impulse in that x direction.
And the total momentum in the y direction initially,
of which there is none in this case,
would have to equal the total momentum in the y direction
finally, if there's no net impulse in the y direction.
So what's an example involving collisions
in two dimensions look like?
Let's say a metal sphere of mass M is traveling
horizontally with five meters per second
when it collides with an identical sphere of mass M
that was at rest.
After the collision, the original sphere has
velocity components of four meters per second and
three meters per second in the x and y directions.
And we want to know, what are the velocity components
of the other sphere right after the collision?
So assuming there were no net forces
in the x or y direction in this case,
then the momentum will be conserved for each direction,
and since the mass of each sphere's the same,
we can simply look at the velocity components.
So if we started with five units of momentum
in the x direction, we have to end with
five units of momentum in the x direction.
So the x component of the second sphere
has to be one meter per second.
And since we started with no momentum in the
vertical direction initially,
we have to end with no momentum vertically.
So if the first sphere has three units of momentum
vertically after the collision, then the second sphere
has to have three units of momentum vertically downward
after the collision, which gives us an answer of D.
What's the center of mass mean?
The center of mass of an object or a system is the point
where that object or system would balance.
And the center of mass is also the point
where you can treat the entire force of gravity as acting.
The way you can solve for the center of mass
is by using this formula.
You multiply each mass by how far that mass is
from the reference point.
If there's no reference point specified,
you get to choose the arbitrary reference point,
which would designate where x equals zero.
You continue adding each mass times its position.
For positions to the left of the reference point,
those would count as negative positions.
And when you're done accounting for every mass
in your system, you divide by the total mass,
which would be all the masses added up, and the number
you get would be the position of the center of mass.
The center of mass is going to have units of meters,
since it's a location.
The location where the system or object would balance,
and the location where you can treat
the entire force of gravity as acting.
Something else that's extremely important to remember
is that the center of mass of a system will not accelerate
unless there's an external force on that system.
In other words, the center of mass of a system
follows Newton's first law.
If the center of mass of a system is at rest,
then even if the masses in that system
exert forces on each other and move around,
the center of mass of that system will stay put
until there's a net external force on the system.
And if the center of mass was initially
moving to the right at some speed,
that center of mass will continue
moving to the right at that speed,
even if the masses are moving in different directions,
until there's a net force on that system.
So what's an example problem involving
center of mass look like?
Lets say a remote control car of mass M
is sitting at rest on a wooden plank, also of mass M,
in the position seen here.
There is friction between the wheels of the car
and the plank, but there's no friction between the plank
and the ice upon which the plank is sitting.
The remote control car is turned on and off.
What would be a possible final position
of the car and the plank?
Now because the car's at rest and the plank is at rest,
that means the center of mass of this system
is also at rest.
And since there's no net force on this system,
the center of mass is going to have to remain at rest.
Where is the center of mass?
Well the car's mass is at three,
the plank's center is at five,
so the center of mass between the car and the plank
would be at the location of four.
So to find the correct solution,
we just need to figure out which one of these
also has the center of mass at four.
Option A has the car at three
and the center of the plank at three.
That'd put the center of mass at three meters,
but that can't be right, the center of mass can't move,
there were no external forces on our system,
and the center of mass started at rest,
so it's got to remain at rest.
For option B, the center of the car is at four,
the center of the plank is at three,
this would put the center of mass somewhere between
three and four, but again, that can't be right.
We need our center of mass to be at the location four.
Option C has the car at six
and the center of the plank at four.
This would put the center of mass of the system at five,
that can't be right.
We need our center of mass at four.
Option D has the car at five,
and the center of the plank at three.
That puts the center of mass at location four,
just like it was before.
So D is a possible solution.