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## Elastic and inelastic collisions

# Elastic and inelastic collisions

AP.PHYS:

CHA‑4.B.1 (EK)

, CHA‑4.B.1.1 (LO)

, CON‑5.A (EU)

, CON‑5.A.2 (EK)

, CON‑5.A.2.1 (LO)

## Video transcript

- [Instructor] Say there's a basketball heading straight toward a scoop of peanut butter chocolate chip ice cream. So these are gonna collide. There's different ways
you could characterize this collision, but one
thing that physicists are almost always interested
in is whether this collision is going to be elastic or inelastic. What does it mean to say
a collision is elastic? Elastic collision is one where the kinetic energy is conserved. And I don't just mean the kinetic energy of one of the objects, I mean the total kinetic
energy of all the objects. So this is where the total kinetic energy of all colliding objects is conserved. And don't forget, people get confused about this word conserved. That's really just a fancy way of saying the total amount of
kinetic energy is constant, i.e. it remains the same value
before and after a collision. And we could put this into
a mathematical statement. If we're clever we could say alright, total kinetic energy conserved, so if we just write down the basketball has some kinetic energy
before the collision, I'm just gonna use the
letter k for kinetic energy, so I'm gonna have kinetic
energy of the basketball. That's gonna be before the collision so we need another subscript. This is gonna get a little messy. I'm gonna have two subscripts: one to denote which
object I'm talking about, the b will be for basketball, and the second letter is gonna represent when I'm talking about it, i.e. this i is gonna represent initial,
like before the collision. So this is the initial kinetic
energy of the basketball, and if we add to that, 'cause we want the total kinetic energy, if we add to that the kinetic energy that the scoop of ice cream had, I'll use s for scoop of
ice cream and initially, this would represent
the total kinetic energy before the collision. And we could do the same
thing for after the collision, we could say that the
basketball's probably gonna be moving after the collision, so the basketball will have
some final kinetic energy, and if we add to that the kinetic energy that the scoop of ice cream
had after the collision, i.e. finally, this here would
be the total kinetic energy after the collision. If the collision is elastic, that means the total
kinetic energy is conserved, that means that this total
initial kinetic energy has to equal this total
final kinetic energy. I could just say that these two are equal if it's an elastic collision. And this is what we mean by
a collision being elastic. It means that the total
kinetic energy is conserved. For an inelastic collision, the total kinetic energy is not conserved, in other words, this
expression doesn't hold. So if I put that over
here, if it's inelastic, what you can say is that the
total initial kinetic energy does not equal the total
final kinetic energy. And for most inelastic collisions the initial total kinetic energy is greater than the final
total kinetic energy. In other words, in an inelastic collision you'll lose some kinetic energy,
some of this kinetic energy gets transformed into
some other kind of energy and that energy is
typically thermal energy. 'Cause think about it. If this ice cream scoop splatters
right into the basketball and the atoms and molecules that make up the ice cream scoop, so this ice cream scoop is made
out of atoms and molecules, delicious atoms and molecules, and they're not masses
connected with springs, but roughly speaking you can think of the solid as masses,
little tiny molecules or atoms connected by springs. It's really electromagnetic forces here and chemical bonds going on, but that's complicated to
just get a nice visual picture of what's happening. Imagine this collision happens. That's gonna cause this atom or molecule to start oscillating more than it was. This one's gonna start
oscillating more than it was. And since these atoms and molecules now have more kinetic energy on their own, this random thermal energy,
the total kinetic energy that this whole ice cream
scoop's gonna have going forward is gonna be less, because
some of that's gonna be distributed randomly amongst
the atoms and molecules in the ice cream scoop. Now, if it's a really
melted ice cream scoop, if the ice cream scoop's not very cold, these springs are not gonna be very stiff, these atoms and molecules
can just slide around however they want, there
might be a lot of energy, a lot of kinetic energy that gets turned into thermal energy. But if you freeze this ice cream scoop, if you take these things
straight out of the deep freezer, then these bonds are
gonna be a lot stiffer and these atoms and molecules are gonna be much more stuck in place
than they were previously. So once this structure becomes more rigid it's harder to transfer
that kinetic energy into these individual atoms and molecules and it'll become more and more elastic. You'll waste less and less kinetic energy to this thermal energy here. And if you take this idea to the extreme, if you instead try to take a steel ball where these bonds between atoms are extremely stiff and
rigid, you start to approach a collision that might
be considered elastic because your final kinetic
energy might be almost the same as your initial kinetic energy. Now, if I were you, I might
be like "hold on a minute." Total kinetic energy is not conserved, but we just said that kinetic
energy in the collision goes into kinetic energy
of these molecules. That's still kinetic energy, right? Thermal energy is still
mostly kinetic energy. And yeah, it's true. Thermal energy is mostly kinetic energy. I mean there could be a
little potential energy and different kinds of
energy in there as well, when you're dealing with thermal energies. But it is mostly kinetic energy. So we should make a distinction. When we say total kinetic
energy is conserved, we mean the total kinetic energy
of that macroscopic object moving in a certain direction. So the speeds, in other words,
that we're talking about and these kinetic energies are the speeds of the
macroscopic objects, right, of the ice cream scoop itself, not of the individual atoms and molecules. In other words, we're not gonna include the random jiggling kinetic energy that these atoms and molecules have in this calculation over here. Otherwise basically every
collision would be elastic 'cause yeah, that
macroscopic kinetic energy turns into microscopic kinetic energy. But up here we're talking
about the macroscopic kinetic energy of that entire object moving in a certain direction. So to make this clear
let's show an example with some numbers here. Let's just say this basketball
and this scoop of ice cream had a certain speed before the collision. So let's say this basketball was going 10 meters per second before the collision and the ice cream scoop
was going, let's say, eight meters per second. And let's say after they collide this basketball's still
moving to the right but it's only moving at
about one meter per second, let's say, and the scoop of ice cream, let's say, gets to backward and it's now going five meters
per second to the right. And I looked up the mass of a basketball, the mass of a basketball
is about 0.65 kilograms. And now with that mass of the basketball I have to pick the right mass over here for my mass of the ice cream 'cause I picked these velocities
just kind of randomly. So, in order to conserve
momentum for this collision, and almost all collisions
should be conserving momentum, the mass of the scoop of ice cream should be about 0.45 kilograms. Now with these numbers in here we can ask: was this collision elastic or inelastic? And one mistake people make is they say, oh well, they bounced
off of each other, right? Because this basketball
is going to the right at only one meter per second and the scoop of ice cream
is going to the right at five meters per second. They must have bounced off of
each other, they separated, doesn't that mean elastic? And no, that doesn't mean elastic. Just because they bounce off of each other does not imply that it's elastic. It works the other way. If it's elastic they do have
to bounce off of each other, but just because it bounces
does not mean it's elastic. So be careful there. Just 'cause they bounce here
does not mean it's elastic. What do we do to check
whether it's elastic? What we do is we check whether
the total kinetic energy was conserved or not. So let's just check. We've got enough numbers
here to figure that out. So I can use the formula
for kinetic energy, which is one half m v squared. And I can find what is
the initial kinetic energy of the basketball, it'd be one
half mass of the basketball times the initial speed of
the basketball, which was 10. So I'm using initial speeds here 'cause I want to find the
initial kinetic energy. And I'm gonna have to add to that, because I want the total kinetic energy I have to add to that the
initial kinetic energy of the scoop of ice cream. So it's gonna be plus another one half times the mass of the scoop of ice cream times its initial speed, which
was eight meters per second. You might say, isn't it negative v? We're gonna square this
anyway so it doesn't matter, so don't forget the square. And if we add all those
up, we get 46.9 Jules of total initial kinetic energy. So is this equal to the final now? Let's just find out the final
amount of kinetic energy. If I take the final
speed of the basketball and use that to find
the final kinetic energy of the basketball, I'd have
one half mass of the basketball times the final speed, is
only one meter per second, and I still square it, and
then I have to add to that the final kinetic energy
of the scoop of ice cream, which is gonna be one half the mass of the scoop of ice
cream times five squared 'cause five was the final speed
of the scoop of ice cream. And if I add all that up,
I get that this equals 5.95 Jules of total final kinetic energy. So is this collision elastic? No way, it's not even close. This initial total kinetic
energy was 46.9 Jules, this final total kinetic
energy was 5.95 Jules, the kinetic energy here was not conserved and because it was not conserved we would consider this
an inelastic collision. But if you're clever, you can
just look at the numbers here. You didn't actually have to
go through all this work. You could just say, hey,
the basketball started with 10 meters per second, it ends with one meter per second. It's definitely got less kinetic
energy than it did before. And this ice cream scoop started with eight meters per second and it ends with five meters per second, it also ends with less kinetic
energy than it did before. So this final kinetic
energy has to be smaller than the total initial kinetic energy. And you can ask: where did that energy go? It goes into the thermal energy
of these molecules and atoms in the objects vibrating
thermally a little more than they did before,
including in the basketball. As well as sound waves
that can get created that also takes away energy, there's lots of places for energy leaks, and in this particular collision
there were a lot of leaks because we lost a good majority of the kinetic energy
that we started with, which made this an inelastic collision. So recapping, for a
collision to be elastic it's not enough to just know it bounces. You have to see if the
total initial kinetic energy is the same as the total
final kinetic energy. If that's the case, it's
an elastic collision, and if that's not the case,
it's an inelastic collision. One last note. Sometimes you'll hear the word
perfectly elastic collision. Well that's redundant. That's just another way to
say an elastic collision. In other words, a collision
where the initial kinetic energy really is equal to the
final kinetic energy. But you'll also sometimes hear about a perfectly inelastic collision. And this is meaningful. This means that the two objects
that collide stick together so if it's perfectly inelastic, this means that they must stick together
and move off as a single unit. In other words, if the scoop of ice cream splattered into the basketball
and then stuck to it, and the two moved off to
the right at some speed, that would be a perfectly
inelastic collision. Now, whether it's elastic or inelastic, momentum is still gonna be
conserved for these collisions. If that collision happens
over a short time interval, there's not enough time
for an external force to cause enough impulse to
impact the momentum greatly. So if it's one of these
instantaneous impacts that happen in collisions,
then the momentum will be conserved for
both elastic collisions and inelastic collisions. Sometimes people get
confused, they're like, wait, I know that energy is only conserved for elastic collisions. Maybe that means that
momentum's only conserved for elastic collisions? But that's not true. Momentum will be conserved
for both inelastic and elastic collisions You might object, you might
be like, wait wait wait. If you're clever, you
might be like, hold on. In these inelastic collisions we're losing all kinds of energy to the
random thermal oscillations in this material. Aren't we also losing momentum
to those random oscillations? I mean movement implies both
kinetic energy and momentum, so why aren't we losing momentum in these inelastic collisions? And the reason is the oscillations of the atoms and molecules
in this material, they're oscillating randomly,
in random directions. This thermal energy gets
distributed in a random way so that the momentum of
the atoms and molecules in that structure cancel out because if you've got momentum
in every single direction, and momentum is a vector,
that equals no momentum, at least no net momentum, because these are all gonna cancel out. This one cancels with this one, this one cancels with that one, that one cancels with that one. So that's why in an inelastic collision there's no loss of total momentum to the microscopic atoms
and molecules of the object, but there is a loss of kinetic energy because kinetic energy is a scalar, kinetic energy has no direction. Kinetic energy can't cancel in this way because it's not a vector. So even though in an inelastic collision you lose kinetic energy to the microscopic atoms and molecules, you don't lose any net momentum to them because all that momentum
just cancels out. And the bulk motion of
these macroscopic objects must maintain the total momentum. And this is wonderful news actually because that means
momentum's gonna be conserved for both elastic and inelastic collisions. It doesn't matter what
kind of collision it is, momentum is gonna be conserved
as long as there is no time for any net external impulse
to act during that collision. So even though energy is only conserved for elastic collisions,
momentum will be conserved for every collision.