mr.p: Good morning, today we are going to
increase our knowledge of tip-to-tail
vector addition by slightly altering
our example involving a moving track
and our slow velocity racer.
Here we go.
Bo: Hey guys.
Billy: Hi Bo.
♫ (lyrics) Flipping Physics ♫
mr.p: Last time we had slow velocity racer
racing parallel to the moving track.
To make things a little more interesting
we're going to turn the track 90 degrees
and increase the width of the moving track.
As we figured out last time, slow velocity racer
can race at 42 millimeters per second
and we are going to have her racing east.
Now, let's take a look at the moving track.
We need to measure the displacement
and time for the moving track to determine
the velocity of the northward moving track
and you can see the track moves 0.60 meters north
or 600 millimeters north in 12.2 seconds.
So, we can use our equation for velocity
to determine the velocity of the track.
Bo, what is the equation for velocity?
Bo: Velocity equals displacement over change in time.
And Billy, what was the displacement of the track?
Billy: 600 millimeters
Bobby: Careful, displacement is vector
and 600 millimeters is just
the magnitude of the displacement.
Bo: Yeah, you need the direction too.
Billy: That's right, vectors have both
magnitude and direction and so the displacement
of the track is 600 millimeters north.
Thanks guys.
Bobby and Bo: You're welcome.
mr.p: So the velocity of the race car
ends up being 600 millimeters north
divided by 12.2 seconds or 49.180
or approximately 49 millimeters per second north.
Now, the way we illustrate the velocity vectors
for the track and slow velocity race car
mathematically are like this.
Just so you know, it should be pretty clear
from the video that these velocities
are not perfectly constant so we're just
going to use two significant digits
on these velocities.
Now I know it's not perfect but it's
good enough for our demonstration today.
Billy: Yeah, two sig figs does make sense.
Bobby: I bet the school's budget
is pretty tight right now.
Bo: Yeah, it's all about the Benjamins baby.
mr.p: Okay, slow velocity racer's velocity vector
is going to be represented by this
capital B with an arrow over the top.
The arrow over the top illustrates
that capital B is a vector.
Sometimes in books you'll see it
in bold face type, you'll also see it
with the arrow over the top.
I can't really do bold face on the board
so I will always be using the arrow.
So the race car has a velocity vector B
with a magnitude of 42 millimeters per second
and a direction of east.
The track is illustrated with
a velocity vector capital A again
with the arrow over the top.
So the track has a magnitude of
49 millimeters per second and a direction of north.
Bobby: Why A and B?
mr.p: It doesn't really matter.
I just picked the first two letters of the alphabet.
Really you can pick whatever letters you would like.
So what we're going to do now
is we're going to figure out what
our slow velocity race car's velocity would be
if she were to race on the track.
So it's going to be vector addition
and I'm going to draw visually
what it looks like on the board now.
So this is vector A for the northward
moving track and this is vector B
for the eastward moving race car.
Notice that vector A is 49 millimeters long
and vector B is 42 millimeters long,
so that they both have a length
which matches the magnitude of those vectors.
This is called, tip-to-tail vector addition
because the tip of vector A is
on the tail of vector B.
Now, let's take a look at what happens
when slow velocity racer races on the track.
So you can see when we add the 49 millimeters
per second north to the 42 millimeters
per second east, we get slow velocity racer's
new velocity which we can illustrate
on the board like this.
And you can see this vector R illustrates
exactly how she ends up moving on the track.
What we are doing here
is finding the results of adding vectors A
and B together and what we get is called,
the resultant vector R.
In this particular case, our resultant vector
is slow velocity race car's resultant velocity
from racing on the track.
Bo, how are we going to find this resultant vector?
Bo: We should just be able to use
A squared plus B squared equals C squared.
mr.p: Yes, we can use the Pythagorean Theorem.
And what is it about this triangle
that tells us we can use
the Pythagorean Theorem, Billy?
Billy: Because north and east are 90 degrees
from one another which makes this a right triangle.
mr.p: Correct, remember we can only use
the Pythagorean Theorem when
you have a right triangle.
So now we can find the resultant vector like this.
Substituting into our equation we get
the resultant vector squared is equal to
vector A squared plus vector B squared
and taking the square root of both sides
we get the resultant vector is equal to
the square root of A squared plus B squared,
substituting in our numbers, we get
that the resultant vector is equal to
the square root of 49 squared plus 42 squared
and what do we get for an answer?
Bo: That should be 64.537 or with two sig figs,
65 millimeters per second.
mr.p: In other words, slow velocity race car's
resultant velocity is 65 millimeters per second.
In other words, slow velocity racer's
resultant velocity vector is 65 millimeters per second.
Bo: Oh, vectors have both magnitude and direction.
Velocity is a vector so we need to figure out
slow velocity racer's direction as well.
mr.p: Correct, we have figured out the magnitude
of slow velocity racer's resultant velocity,
however, we also need to figure out
the direction in order to figure out
the resultant velocity vector of our race car.
Billy, how are we going to figure out
the direction of our resultant vector?
In other words, how are we going to
figure out this angle theta?
Billy: SOHCAHTOA.
mr.p: SOHCAHTOA.
All: SOHCAHTOA.
mr.p: Yes Billy, we have a right triangle
so we can use SOHCAHTOA, but how?
Billy: Let's use TOA so tangent
equals opposite over adjacent.
(cries and bang head)
mr.p: Why do they always forget?
Billy: Oh sorry, I forgot theta.
Tangent theta equals opposite over adjacent.
Bo: It's not funny, I forget theta too.
mr.p: How could you forget poor theta?
Don't you worry, you are not alone.
Often students forget theta.
It makes me sad every time.
Let's move on.
Opposite of our angle theta is the vector B
and adjacent to our angle theta is vector A.
B has a magnitude of 42,
A has a magnitude of 49.
Taking the inverse tangent of both sides
we get that theta is equal to the inverse tangent
of 42 over 49 which works out to be.
Bo: That's 40.601 or 41 with two sig figs.
mr.p:: So we independently figured out
the magnitude and the angle for
the resultant velocity of our race car.
Which means the resultant velocity vector
works out to be with two significant digits,
65 millimeters per second at an angle
of 41 degrees east of north.
Now, many of you won't understand
where I got east of north from
and that's absolutely okay at this point.
I'm going to go through in our next lesson
how to work with cardinal directions in physics.
So, let's take a moment and review
exactly what this means.
When slow velocity racer races
at 42 millimeters per second east
on a track that is moving at 49 millimeters
per second north, she will actually
be traveling at 65 millimeters at an angle
of 41 degrees east of north because
49 millimeters per second north
plus 42 millimeters per second east,
equals 65 millimeters per second
at an angle of 41 degrees east of north.
Bobby: Hold up Mr. P, I think maybe
I'm confused or I'm missing something.
How did we add 49 and 42 and get 65?
Shouldn't we get 91?
mr.p: Awesome question, I love this question.
Thank you.
How can 49 plus 42 equal 65?
Bobby: Exactly.
mr.p: The answer to your question is that
your question is wrong.
Billy: How can a question be wrong?
mr.p: Your question is wrong because
49 plus 42 doesn't equal 65 but we didn't
add 49 plus 42.
What did we add Bo?
Bo: Oh yes, the issue is that we added vectors
not scalars so it doesn't work.
It doesn't make any sense if you
just drop the direction.
Billy: Yeah, so 49 millimeters per second north
plus 42 millimeters per second east
equals 65 millimeters at an angle
of 41 degrees east of north.
mr.p: Correct, we didn't add scalars here,
we added vectors and if you don't include
the direction, it just doesn't make sense.
49 plus 42 definitely does not equal 65.
However, 49 millimeters per second north
plus 42 millimeters per second east
does equal, with two sig figs,
65 millimeters per second at an angle
of 41 degrees east of north.
Are there any more questions?
Bobby: Nope.
Billy: No.
Bo: Not if we're going to go over that
weird east of north thing next time.
mr.p: Yes, absolutely, we are going to go over
cardinal directions in our next lesson.
Let's do a quick review.
We did tip-to-tail vector addition.
You can illustrate that visually on the board
using arrows of the correct length.
We did that using letters with an arrow
over the top to illustrate that they are vectors.
Because we had a right triangle, we could use
the Pythagorean Theorem and SOHCAHTOA
to figure out both the magnitude
and the direction of our resultant velocity vectors.
Vectors, vectors, vectors.
Thank you very much for learning with me today.
I enjoyed learning with you.