- Good morning.
Today we are going to review
momentum, impulse, collisions, and center of mass
as a part of the AP Physics C Mechanics curriculum.
(theme song) ♫ Flipping Physics
- Bo, what is the equation for momentum?
- Momentum equals mass times velocity.
- We should point out, momentum is a lower-case p,
power is an upper-case P.
- And momentum and velocity are vectors.
- And units for momentum are kilograms, meters per second.
- And the units have no special name,
they are just kilogram meters per second.
Unlike newtons, which are
kilograms meters per second squared.
- Okay.
- Billy, what is Newton's second law in terms of momentum?
- The sum of the forces acting on an object
or system of objects equals the derivative
of the momentum of that object or system of objects,
with respect to time.
- Don't forget, force and momentum are vectors.
- Yeah.
- Because momentum equals mass times velocity,
we can use the product rule to get.
Net force equals the derivative of mass
with respect to time times velocity,
plus mass times the derivative of velocity
with respect to time.
And, we know the derivative of velocity with respect to time
equals acceleration, therefore, Bobby,
when we use the net force equals mass times acceleration
as Newton's second law, what were we assuming?
- Well, that would mean the derivative of mass
with respect to time was zero.
- [mr.p] Sure, Bobby, but what does
that really mean about the object?
- Oh, that means the mass of the object
is not changing, I get it.
Net force equals mass times acceleration
assumes the mass of the object is constant.
- Billy, when can we use conservation of momentum?
- We can use conservation of momentum
when all the forces are internal to the system.
- [mr.p] That is correct, but can you prove it?
- Sure, uh, if all the forces are internal
to the system, then the sum of the external forces
acting on the system equals zero,
and the derivative of the momentum
of the system with respect to time equals zero.
And if the derivative of the momentum of the system
with respect to time equals zero,
then the momentum of the system is not changing
as a function of time,
so the sum of the initial momenta
equals the sum of the final momenta.
- It is also useful to realize
momentum is conserved during all collisions and explosions.
- Because all the forces are internal to the system
during collisions and explosions.
- Now let's derive the equation for impulse,
which is often called the impulse momentum theorem.
We start by looking at the net force acting on an object,
and using the equation we were just working with,
we can rearrange that to get
net force times dt is equal to dp.
And then we can take the definite integral
of the whole equation.
The limits when taking the integral with respect to time
go from time initial to time final.
And the limits when taking the integral
with respect to momentum go from momentum initial,
to momentum final.
The definite integral of one with respect to momentum
is momentum final, minus momentum initial,
which is change in momentum.
In other words, the change in momentum
equals the definite integral
from time initial to time final
of the net force with respect to time.
Which is defined as impulse,
and the symbol for impulse is a capital J,
and impulse, just like momentum, is a vector.
Because impulse is force times time,
the units for impulse are Newton's seconds.
- [Billy] Mr. P?
- And, yes, Billy,
the units for impulse, Newton's seconds,
are equivalent to the units for momentum,
kilogram meters per second.
- [Billy] I was going to ask that.
- Yeah, I know.
Remember, integrals are the area under the curve.
Therefore, impulse is the area between
the net force as a function of time curve,
and the horizontal time axis,
where area above the time axis is positive,
and the area below the time axis is negative.
Also,
do not confuse impulse with work.
Work is the definite integral of
force with respect to position, not time.
The two equations are very similar to one another.
Okay, Bo, what is the impulse approximation?
- The impulse approximation
is where we can say the force of impact
during the collision is so large that it
overshadows all the other forces,
and therefore the net force is approximately equal
to the force of impact during the collision.
Uh, that is true with large forces of impact,
which usually last for a short time interval.
- Sadly, impact force and impulse often get confused
with one another, perhaps it's because
they both start with 'imp', I don't know.
If we can consider the force of impact
to be constant, or we have the average force of impact,
the impulse is equal to the average force of impact
times the change in time.
This gives us a rectangle with the same area
as the definite integral impulse equation.
Now let's review the types of collisions.
Bobby, please explain this table.
- This table lists two types of collisions,
elastic and inelastic.
Momentum is conserved during all types of collisions,
so the difference between elastic and inelastic collisions
is whether kinetic energy is conserved.
And kinetic energy is conserved during elastic collisions,
and kinetic energy is not conserved
during inelastic collisions.
- What about perfectly inelastic collisions?
- Yes, a perfectly inelastic collision
is where the two objects run into one another
and stick together afterwards.
I've also seen a perfectly inelastic collision
referred to as a completely or totally inelastic collision.
Be aware that collisions between hard spheres
are considered to be elastic collisions generally,
in physics classes.
And in the real world, most collisions are actually
inelastic, where some kinetic energy is converted
to heat and sound during the collisions.
Okay, next let's talk about center of mass.
The position of the center of mass
of a system of particles is equal to the sum
of the mass of each particle, multiplied by
the location of each particle, all divided by
the total mass of the system.
In other words, if we are solving for the center of mass
of a system of particles in the X direction,
that is equal to the mass of object one,
multiplied by the X position of object one
relative to a zero reference line,
which often is, but does not have to be, the origin,
plus the mass of object two,
times the location of object two,
again in the X direction relative
to the zero reference line,
plus however many objects we have,
we repeat that that many times,
all divided by the total mass of the system of particles.
The velocity of the center of mass of a system of particles
is equal to the derivative of the position
of the center of mass of the system of particles
with respect to time.
And because the mass of the system of particles
does not change, and the derivative of position
with respect to time is velocity,
the equation for the velocity of the center of mass
of the system of particles is almost identical
to the equation for the position of the center of mass
of a system of particles, you simply replace
position with velocity.
And, we can do the same thing to determine
the acceleration of the center of mass
of a system of particles.
Because the derivative of velocity with respect to time
is acceleration, the equation for the acceleration
of the center of mass of a system of particles
is almost identical to the equation for the velocity
of the center of mass of a system of particles.
You simply replace velocity with acceleration.
Now, realize the only one of these three equations
which is on the AP equation sheet,
which you're going to get on the exam,
is the equation for the position of the center of mass
of a system of particles.
The equations for velocity and acceleration
of the center of mass of a system of particles
are not on the AP equation sheet.
Now let's talk about finding the center of mass
of a rigid system of particles,
or what is more commonly called a rigid object with shape.
The position of the center of mass of a rigid object
with shape, is equal to one over the total mass
of the object, times the integral,
with respect to mass, of the location of all
of those infinitesimally small pieces of that object,
which are called dm, relative to a zero reference line.
- Could you please say that again?
- Yeah, yeah, please. Please.
- What is 'r'?
- Uh, sure, I'll start with 'r'.
Now, 'r' is a position vector, just like we used
in kinematics, and it describes the position of an object
in one, two or three dimensional space.
And, so here it's describing the location
of all of these dms.
And when we're figuring out the position center of mass,
we usually only determine that in one dimensional space,
so we usually replace it with something
we're more familiar with, like X.
So in this case, X is defined as,
in the same way it was for a system of particles,
it is the location of all of the different pieces
of the system, in this case,
the system is the object, so we break the object
into an infinitely large number of
infinitesimally small pieces, dm.
- That's right, that's what an integral is.
- Exactly, that is what an integral is.
And this particular integral is not on
your AP equation sheet, I don't know why,
it's just not there.
- [Bo] That does not help me out at all.
- I know, um. Okay, so.
On to more things which are not on the AP equation sheet.
Volumetric mass density equals the mass of the object,
divided by its volume.
The symbol for volumetric mass density is
the lower-case Greek letter rho,
and just so you know, I put a slash through my V
for volume, to distinguish volume from velocity.
Surface mass density is the mass of the object
divided by its surface, and the symbol
for surface mass density is the
lower-case Greek letter sigma.
And linear mass density equals the mass of the object,
divided by its length.
And the symbol for linear mass density
is the lower-case Greek letter lambda.
- Did he really say none of those are on the equation sheet?
- Yep. - Oh.
- He did.
- These equations for density tend to be helpful
when determining the center of mass
of rigid objects with shape,
although I will admit that sigma,
the surface mass density, I have yet to see
be used in AP Physics C Mechanics,
however, in AP Physics C, electricity and magnetism,
the surface charge density is used,
and I thought y'all should start getting used it.
- Thank you. - Sure.
- That completes my review of momentum, impulse,
collisions, and center of mass.
Next feel free to enjoy my review of rotational kinematics
for AP Physics C Mechanics,
or you can visit my AP Physics C review webpage.
Thank you very much for learning with me today,
I enjoyed learning with you.