- Good morning.
Bo, could you please read the problem,
and Bobby, could you please translate?
♫ Flipping Physics ♫
- On a level surface, a street hockey puck
is given an initial velocity of -3.2 meters per second
and slides to a stop.
- Please stop.
Velocity initial equals -3.2 meters per second,
and velocity final equals zero.
- Bo, please continue.
- If the coefficient of kinetic friction between the puck
and the surface is 0.60, how far did the puck slide?
- Mu k equals 0.60,
and delta x equals question mark;
although, it's just how far, so just the magnitude
of the displacement.
- I want my two dollars.
- Correct.
Bobby, why is the initial velocity of the puck negative?
- Because the puck is moving to the left?
- Yes, the initial velocity is negative,
because the puck is moving to the left.
Now we have solved problems like this before
using Newton's Second Law and the
Uniformly Accelerated Motion equations;
however, today we are going to use the equation
work due to friction equals change in mechanical energy
to solve this problem.
Billy, how do we know we can use this equation
to solve this problem?
- Because there is work done by the force of friction;
however, there is no work done by a force applied.
- Yes, we can use the work due to friction
equals change in mechanical energy equation
when there is work done by friction,
however, there is no work done by the force applied.
Bo, could you please expand the work
due to friction equation?
- Okay.
Work due to friction equals
force of kinetic friction times displacement
times cosine theta, and the change in mechanical energy
equals mechanical energy final
minus mechanical energy initial.
- Billy, before we can use this equation--
- You cut your hair.
- [Bo] Nope - Yes you did, Bo!
You cut your hair!
- No, I did not.
- Bo did not cut his hair.
Bo got his hair cut.
- (laughs) So did I.
Can we get back to the physics please?
(laughing) Billy, before we can use this equation,
what do we need to do?
- We need to identify the locations
of the initial and final points
and the horizontal zero line.
- Then could you all please identify those?
- Okay, let's set the initial point
where the puck leaves your hand.
- And the final point where the puck stops.
- And let's set the horizontal zero line
at the height of the center of mass of the puck.
- Great, now we can work with the equation.
Bobby, what can we do with the left-hand side?
- We can substitute the coefficient of kinetic friction
times force normal for the force of kinetic friction,
and the displacement of puck is to the left,
and the force of kinetic friction is to the right, and
the angle between those two directions is 180 degrees,
so theta equals 180 degrees.
- All right, now, the right-hand side of this equation
is very similar to conservation of mechanical energy.
And we have worked a lot by now
with conservation of mechanical energy.
We've even taken a quiz on conservation
of mechanical energy, which means, at this point,
I'm going to say we can graduate.
And rather than listing all the energies
which could possibly be there,
we are only going to list the energies
which actually are there.
Is everybody okay with that?
- [Billy] I'm okay with that. - [Bo] About time.
- Can we still list all the energies
if we feel more comfortable doing it that way?
- Absolutely, Bobby.
If you would like to, you may still list all the energies
which could possibly be there, but you do not have to.
Billy, could you please tell me, initially and finally,
which energies are there and are not there, and why?
- Okay, there is no spring, so no elastic potential energy,
initial or final.
The height of the puck is zero, both initially and finally,
so there is no gravitational potential energy at all.
The final velocity of the puck is zero,
so the kinetic energy final is zero;
however, the initial velocity of the puck is not zero,
so the initial kinetic energy is not zero.
Is that really true?
There is no final mechanical energy,
and the only energy initial is the kinetic energy?
- Yes, that is true, there is zero final mechanical energy,
and the only initial mechanical energy is kinetic energy,
so zero minus one half mass times velocity initial squared.
But where did all that initial kinetic energy go?
(sliding puck)
- Oh you can hear it.
Some of the kinetic energy becomes sound.
- And some of the kinetic energy becomes heat.
The puck is warmer after sliding along the table,
just like your hands heat up when you rub them together.
- [mr.p] Yes, all that initial kinetic energy is dissipated
as heat and sound.
So let's put that equation in our equation holster.
And Bo, please go through the equation from left to right,
and tell me which variables we do and do not know.
- We know the coefficient of kinetic friction.
We don't know the force normal or the displacement.
The cosine of 180 degrees is negative one.
We don't know the mass of the puck.
We do know the velocity initial.
Oh (laughs), we need to draw a free body diagram
and sum the forces to solve for the force normal, right?
- Correct.
Bobby, could you please do that?
- Sure.
The force normal is up,
the force of gravity is down,
the force of kinetic friction is to the right,
and the force applied by your hand is to the left.
- [Mr. P] So you're talking about the force applied
by your hand?
- Right, sorry, your hand is not touching the puck
while it is sliding to a stop, so there is no force applied
in the free body diagram, sorry.
- [Mr. P] Bobby, you don't need to apologize.
That happens all the time.
- Okay, we can sum the forces in the y direction.
The net force in the y direction equals
force normal minus force of gravity,
and the net force in the y direction also equals
mass times acceleration in the y direction.
Because the puck does not move up or down,
the acceleration in the y direction is zero.
Therefore, the force normal equals the force of gravity,
which equals mass times the acceleration due to gravity.
- Bo, please keep going.
- Using the equation from the equation holster,
we can substitute mass times acceleration due to gravity
for force normal, and, wait for it,
everybody brought negative mass to the party.
(techno music)
♫ Everybody brought mass, mass ♫
- (laughs) And we are left with the coefficient
of kinetic friction times the acceleration due to gravity
times displacement is equal to one half times
the initial velocity squared.
Bo, please keep going.
- Divide both sides by acceleration due to gravity
and the coefficient of kinetic friction
to get displacement equals velocity initial squared
divided by the quantity two times acceleration
due to gravity times the coefficient of kinetic friction.
With numbers, that is the quantity negative 3.2 squared
divided by the quantity two times 9.81
times 0.60,
which works out to be
0.869861,
or with two significant digits, 0.87 meters.
- Yes, thank you very much for--
- Wait a second, the puck moves to the left.
The displacement should be negative.
Our answer should be negative.
I give up, why does this always happen?
Why can't we just get the right answer?
- Bobby, chill out.
We got the right answer.
- That is true, Bo, please explain.
- When using the work equation,
you only use magnitudes of the force and the displacement.
So if you solve for the force or the displacement
in the work equation, you will only get
the magnitude of that variable.
- Right, so our answer is the magnitude
of the displacement of the puck,
which is the answer to how far the puck slid.
Our answer is correct.
- Absolutely.
Thank you very much for learning with me today.
I enjoyed learning with you.
- [Bobby] Does anybody know where my pencil is?