Bo: Hey, guys.
Billy: Hey, Bo. Bobby: Hi, Bo.
♫ (lyrics) Flipping Physics ♫
Mr. P.: Ladies and gentlepeople,
the bell has rung,
therefore class has begun,
therefore you should be seated in your seat
ready and excited to do an
introductory uniformly accelerated motion
or UAM problem.
Bobby: Absolutely.
Billy: Yes.
Bo: Oh, boy.
Mr. P.: All right, let's get right
to the example problem.
We're going to actually take
a more detailed look at one of the examples
from last time, the bike braking example.
Bo, could you please read the problem?
Bo: Sure.
Mr. P. is riding his bike at 22.9 kilometers
per hour when he applies the brakes
causing the bike to slow down
with a constant acceleration.
After 1.01 seconds he has traveled 4.00 meters.
Part (a) What was his acceleration,
and Part (b) what was his final speed?
Mr. P.: All right, it's time to translate
that problem to physics.
Bo, could you please read the problem again,
and Bobby, could you please translate?
Bo: Mr. P. is riding his bike at 22.9 kilometers
per hour when he applies the brakes
causing the bike to slow down
with a constant acceleration.
Bobby: Please stop.
Velocity initial equals 22.9 kilometers per hour,
and we know we use the UAM equations
because it says the acceleration is constant.
Mr. P.: Yes. Very nice, Bobby.
It is important that you are able
to identify when you can use
the uniformly accelerated motion equations,
in other words, when the object is in
uniformly accelerated motion.
It does say in the problem
that it is a constant acceleration; therefore,
we can use the UAM equations.
Now I know it's in the title of the video
so certainly you can use it this time,
but it is important that you're able to identify
when you can use the UAM equations.
Please don't tell me "I have 3
of the UAM variables, therefore it must
be uniformly accelerated motion."
That is not always true.
Bo: After 1.1, no, after 1.01 seconds
he has traveled 4.00 meters.
Part (a) What was his acceleration,
and Part (b) What was his final speed?
Bobby: Okay. Change in time is 1.01 seconds.
The displacement is the 4.00 meters.
For Part (a), acceleration equals question mark,
and Part (b), velocity final equals question mark.
Billy: Hold up.
I thought the question asked for the final speed
not the final velocity.
Mr P.: That was very perspicacious, Billy.
Billy: Thanks.
Mr. P.: You are welcome.
When an object is moving in a straight line
as it is in this problem,
then at the speed and a specific time or location,
and the velocity at that same time or location
are going to have the same magnitude.
So yes, we can use the UAM equations.
We're just going to get the magnitude
of those variables.
Again, that's because in this example
we're moving in a straight line.
For example, we won't actually
figure out the final velocity.
We're going to figure out the final speed,
and we won't actually figure out the acceleration,
we'll figure out the magnitude of the acceleration,
the acceleration without a direction.
Now it probably should be more clear
from the problem statement,
but it is pretty typical of problems
of this ilk that they pretty much ignore
the direction.
I wanted to model that in this problem
because you are going to run into problems like that
in your physics career.
Bobby: Oh, so the 4.00 meters isn't actually
the displacement.
It's the distance traveled.
However, because you moved in a straight line
they're both the same thing right?
Mr. P.: Correct.
It's the magnitude of the displacement
that we have here of 4 meters,
not the displacement itself
because it doesn't have a direction.
You're correct there.
Billy, how would you like to solve the problem?
Billy: Well, we have 3 UAM variables
so we just need to pick the correct UAM equation
in order to find the other 2 UAM variables.
We know the initial velocity, the displacement,
and the change in time,
and we're trying to find the acceleration.
There should only be 1 equation
that has those 4 variables in it.
It's the displacement equals velocity initial
times the change in time plus 1/2
times the acceleration times
the change in time squared.
Mr. P.: Yes, this is the only UAM equation
we can use in this case.
Now, many of you are going to feel compelled
at this point to plug in numbers,
and I would not.
I would actually suggest that you first start
by rearranging the equation to solve for
the acceleration please.
Billy: Why? It's probably easier to plug
in numbers at this point.
Mr. P.: Actually, I would consider which one
is easier to be debatable,
and we're not going to have the debate.
It doesn't seem worth our while.
My point is more that in the long run
we're going to have to rearrange a lot
of equations in physics,
and it's best to get used to that right now
so it's easier when we get farther into the class.
Bo, could you please rearrange this equation
to solve for the acceleration?
Bo: Well, we should start by subtracting
velocity initial times
change in time from both sides.
That leaves us with change in position
minus initial velocity
times the change in time equals 1/2 acceleration
times change in time squared.
Then we can divide both sides
by 1/2 change in time squared,
and we are left with moving the acceleration
over to the left with acceleration is equal
to displacement minus velocity initial
times change in time.
That quantity divided by 1/2
of the change in time squared.
Mr. P.: Very nice, Bo.
You subtract the velocity initial
times the change in time from both sides,
and then you divide both sides by 1/2
times the change in time squared.
You end up with the acceleration equals
the displacement minus the velocity initial
times the change in time, that whole quantity
divided by 1/2 times the change in time squared.
Now, it doesn't really matter on which side
we have the variables.
We know that they're both equal
so we could say this is acceleration equals
the displacement minus velocity initial
times the change in time,
that whole quantity divided by 1/2 times
the change in time squared.
Bobby, what would you like to do next?
Bobby: Well, we have all the numbers
so let's just plug them in.
Billy: Careful. Dimensions are your ...
Bo: Friends.
Billy: You need to make sure
they play well together.
Bobby: We need to convert to meters and seconds
before we plug them into the equation.
Mr. P.: Absolutely.
Bobby: The only thing not in meters and seconds
is the initial velocity
so multiply the 22.9 kilometers per hour
by 1 hour over 3,600 seconds
to cancel out the hours,
and multiply by 1,000 meters over 1 kilometer
to cancel out the kilometers,
and we're left with ...
Hold up. Give me a second.
6.36 with a repeating 1 at the end
meters per second.
Mr. P.: Yes. You need to make sure that your
dimensions play well togther.
You should never put kilometers and hours
on the same team as meters and seconds.
it will just not go well.
Now that we have all of our numbers,
we can simply plug those in.
And Bo, what do we get for an answer?
Bo: -4.75389, or with 3 sig figs, -4.75.
Billy: Dimensions are your friend.
Bo: -4.75 meters per second squared.
Mr. P.: Great. So for an answer for Part (a)
we get -4.75 meters per second squared.
That's rounded to 3 sig figs
because the least number of significant digits
from our givens was 3.
Bobby, could you please work on Part (b)?
Bobby: Let's see.
We could use the equation velocity final squared
equals velocity initial squared
plus 2 times the accleration
times the change in position.
We're looking for the final velocity
so we just take the square root
of the whole equation,
and we get that the final velocity
is equal to the square root
of the initial velocity squared
plus 2 times the acceleration times
the change in position.
We know all the numbers,
and they're all in meters and seconds
so they'll play well together
so we could just plug in the numbers.
Velocity final equals the square root of 6.36
with a repeating 1, the 6.361 squared
plus 2 times the -4.75 times 4,
and we're going to get ...
Billy: 1.56963 which rounds to 1.57 meters
per second with 3 sig figs.
Mr. P.: Sadly, we have the wrong answer.
Billy: I used the rounded answer for Part (a).
Bo: That's right.
Never use a rounded answer.
Always use the unrounded answer
with enough sig figs.
Bobby: Or your answer will be wrong.
Is that actually true?
Mr. P.: Yes, that's absolutely true.
Let's take a look at what happens
when we actually write in the correct numbers.
Notice that when we use the unrounded answer
for Part (a) in Part (b),
we end up with our final velocity of 1.55968
which rounds to 1.56 meters per second
which is clearly not 1.57 meters per second.
Please always use the unrounded answer
when you're working through a problem.
Bo: Does it really matter?
Mr. P.: Yes.
Bo: Okay. It just seems pedantic to me.
Mr. P.: Uh, nice use of the word pedantic;
however, yes it absolutely makes a difference.
In physics you're either right or you're wrong,
and this answer is wrong.
I guess in physics we all have to be pedants.
Oh, and speaking of pedantic,
I do want to remind everybody
that what we found here was the final speed
not actually the final velocity
because we don't have a direction.
Yup, Bobby, what's your question?
Bobby; For Part (b), wouldn't it have been easier
to use the equation velocity final equals
velocity initial plus acceleration times
the change in time instead?
Mr. P.: Actually, there are 3 different equations
we could have used when we got to this point.
The reason for that is because we actually know
4 of the UAM variables.
There are actually then 3 different equations
we could use to solve for the final velocity.
Let me walk through that real quick.
Again, there are 3 different equations
we could have used to solve for the final velocity
because we knew 4 of the UAM variables
at that point.
Which 1 is the easiest is probably quite pedantic.
Any more questions at this point?
Billy: Yes. If I convert 1.55968 meters per second
to kilometers per hour,
I get about 5.6 kilometers per hour.
Why didn't the speedometer on your bike
say 5.6 kilometers per hour?
Mr. P.: Wow. You guys are really paying attention.
That's awesome.
It turns out that the speedometer actually
only takes data 1 time every turn of the wheel.
So during the 1.01 seconds,
the wheel only makes around roughly 2 1/2 turns.
Therefore, it doesn't take enough data
to have registered enough change
in the speed of the bike.
Any more questions?
Bobby: No. Bo: Nope.
Billy: No, but thanks for asking.
Mr. P.: In that case,
I've enjoyed learning with you today.
I hope you've enjoyed learning with me.
Voiceover: Lecture notes are available
at FlippingPhysics.com.
Please enjoy lecture notes responsibly.