- Good morning!
Let's do an elastic collision problem.
♫ Flipping Physics ♫
But first, it's hat, wig, vest, and sword day.
- [Group] Hat, wig, vest, and sword day!
- I don't get it.
Why isn't it just hat day?
- Last year on hat day, everybody just wore a baseball cap.
- It was pretty boring.
- Hence hat, wig, vest, and sword.
- Oh, okay.
- Billy, could you please read the problem?
And, Bo, could you please translate?
- Cart 1 has a mass of 2m and cart 2 has a mass of m.
- Pause.
I guess mass 1 equals 2m
and mass 2 equals m.
- That's fine, Bo.
Billy.
- Cart 2 is initially at rest.
Cart 1 is moving at 40.9 centimeters per second
when it collides elastically with cart 2.
- Pause.
Velocity of cart 2 initial is zero.
Velocity of cart 1 initial is 40.9 centimeters per second.
- Billy.
- If the speed of cart 1 after the collision
is 13.4 centimeters per second,
what is the speed of cart 2 after the collision.
- Velocity of cart 1 final is 13.4 centimeters per second.
And the velocity of cart 2 final equals question mark.
- Thank you.
Now let's take a look at the demonstration.
The way I calculated the velocities
is by measuring the positions of the carts
as a function of time, and calculating the slopes
just before and after the collision.
Also, the question uses speed, and we have used velocity.
We could do this because the carts are moving
at constant velocities, and therefore the magnitudes
of the cart velocities are the same as their speeds.
Bo, how should we solve for the final velocity
of cart 2?
- Well, we know momentum is conserved during collisions,
so the sum of the initial momenta equals the sum
of the final momenta.
This means mass 1 times velocity 1 initial
plus mass 2 times velocity 2 initial
equals mass 1 times velocity 1 final
plus mass 2 times velocity 2 final.
We can substitute in 2m for mass 1.
40.9 for velocity 1 initial,
m for mass 2, 0 for velocity 2 initial,
and 13.4 for velocity 1 final.
Oh, hey, everybody brought mass to the party.
♫ Everybody brought mass
- Bo, please keep going.
- We can subtract 2 times 13.4 from both sides
to solve for velocity 2 final,
and we get 55 centimeters per second.
- But shouldn't we only have one significant digit
on our answer, because the mass of cart 1 is 2 times m
and 2 only has one significant digit?
- That's a good question, Billy.
So, generally when a given value
is a multiple of another given value,
then we treat that multiple as an exact number.
For example, that 2, we treat as an exact number.
Therefore, we should have three significant figures
on our answer.
So, our final speed of cart 2
is 55.0 centimeters per second.
Or, if we were to give the final velocity of cart 2,
it would be 55.0 centimeters per second to the right.
Now, we can look at the full graph
of position as a function of time for both carts.
The slope of cart two after the collision,
I measure to be 52.8 centimeters per second.
So, 52.8 centimeters per second is our observed value
for the velocity of cart 2 after the collision.
We can use the relative error formula now
to solve for the error in our measurement.
Relative error equals observed value
minus accepted value, divided by accepted value times 100.
And we end up getting negative 4.00% error
for our observed velocity.
- [Bobby] Mr. P.
- Yes, Bobby?
- This is an elastic collision,
so kinetic energy should be conserved.
Can we prove that?
- We don't know the mass of either cart,
so I don't think we can prove kinetic energy
is conserved, can we?
- You know what?
Actually, we can.
We know the sum of the initial kinetic energies
should be equal to the sum of the final kinetic energies.
Therefore, the ratio of the total kinetic energy final
over the totally kinetic energy initial
should be equal to one.
The total kinetic energy initial equals the sum
of the kinetic energies of both carts.
In other words, 1/2 times mass 1 times the velocity
of cart 1 initial squared plus 1/2 times mass 2
times the velocity of cart 2 initial squared.
Which equals 1/2 times 2m times 40.9 squared
plus 1/2 times m times 0 squared.
Which equals 1,672.81 times mass.
And we can do the same thing
with the sum of the kinetic energy final
using the 52.8 centimeters per second,
the observed value for the final velocity of cart 2.
We get 1573.48 times mass
for the total kinetic energy final.
And now we can go back to the ratio of kinetic energies.
With the net kinetic energy final
divided by the kinetic energy initial,
you can see that the mass cancels out
and we are left with the number 0.94062,
which we can multiply by 100 to express as a percent.
In other words, 94.1% of the kinetic energy
that existed before the collision,
remains after the collision.
So, class, was kinetic energy conserved?
- No.
- Yes.
- Maybe.
- Yeah, I'm not really sure either.
I would say it's pretty close.
But if you really want to know,
you need to take more trials.
Thank you very much.
- Hold up a second, Mr. P.
I take umbrage with something you just did.
- Okay, Mr. Becke, go ahead.
Take umbrage.
- Well, take a look at cart 1.
It has kinetic energy of 1,672.81m.
That cart looks to be about 517g or 0.517kg.
So, that would be a kinetic energy of 864.84 joules,
because kinetic energy is in joules.
But that's not right, because joules
are newtons times meters,
or kilograms times meters squared over seconds squared.
And you used centimeters per second.
I think you should have converted your velocities
to meters per second.
- But wouldn't you agree that in this particular case
that it does not matter because we take the ratio,
and the units cancel out?
- I will agree to that, if you agree
that it is not a good idea to get into the habit
of using centimeters when determining energy.
- Ah, but we could determine the energy in ergs,
or grams times centimeters squared
over seconds squared.
- Ah, yes.
Ergs.
And have you ever mentioned ergs before?
- No.
- I rest my case.
It is a bad idea to have determined energy
using centimeters.
- I conceed.
You are correct, sir.
- Thank you.
May I?
- You may.
- Thank you very much for learning with us today.
We enjoyed learning with you.
- Thank you, Mr. Becke.
This was fun.
- You're welcome.