mr.p: Good morning.
Today we're going to use a non-terrestrial viewpoint
to give an introduction to relative motion
by way of a one-dimensional example.
Bo: Hey guys.
Billy: Hey Bo!
Bobby: Hi Bo.
♫ Flipping Physics ♫
mr.p: Up to this point all velocities have been
relative to the ground, also known as the Earth.
And unless otherwise stated,
all velocities will continue to be relative to the Earth.
But in today's problem we're going to go through
some velocities relative to other items.
But we're going to start with something familiar,
the velocities of two cars relative to the Earth.
The velocity of the Prius on the left is 13 mi/hr East,
and the velocity of the minivan on the right
is 24 mi/hr, also East.
Both velocities are relative to the Earth.
Now, to better understand this example
let's move our camera into the sky.
As you can see in this video taken from above
using a quadcopter drone that was flown by
Aaron Fown of FirstUAV.
Thank you very much Aaron,
it was great of you to donate your time.
The velocity of the minivan relative to the Earth
is 24 mi/hr East.
And the velocity of the Prius with respect to the Earth
is again, 13 mi/hr East.
In this video, the quadcopter
is stationary relative to the Earth,
so you see the velocities of each car
relative to the Earth which is, again,
what you are most familiar with.
Now let's take a moment to look at
how those subscripts work.
V with a subscript of m and then E
means the velocity of the minivan
with respect to the Earth.
Bo, what then does V with a subscript
of p and then E mean?
Bo: The velocity of the Prius with respect to the Earth.
mr.p: Correct.
Now let's solve for the velocity of the minivan
with respect to the Prius.
Bobby, how do we write that on the board?
Bobby: The velocity of the minivan with respect to the Prius
should be V sub m and then p.
mr.p: Yes, we are solving for the velocity
of the minivan with respect to the Prius.
In other words, when the Prius driver
looks out their window, we're solving
for the velocity at which they would
measure the minivan to be moving.
[Billy] Mr. P?
mr.p: Yes Billy?
Billy: Doesn't that just work out to be 24 minus 13,
or 11 mi/hr East?
Can it really be that simple?
mr.p: Actually, yes Billy, it is that simple.
And there's a reason we're going through
this simple problem.
What we're going to do is we're going to take
this simple problem and make it more complicated
by doing every single step.
And when you understand every single step,
it will actually make more complicated problems easier.
Billy: Good!
mr.p: But before we do, let's take a look at
what the velocity of the minivan
with respect to the Prius looks like.
As you can see in the video,
when we fly the camera directly over the Prius,
we can see what the world looks like relative to the Prius.
And you can see that relative to the Prius,
the minivan is moving at 11 mi/hr.
Now, let's compare that to the video
that is stationary relative to the Earth.
You can see that relative to the Earth,
the minivan appears to be moving much more quickly.
While relative to the Prius,
the minivan appears to be moving more slowly.
And this is what you see from inside the Prius.
The minivan appears to pass the Prius
moving at 11 mi/hr.
Let's get back to solving the problem.
We'll start with the vectors.
I've drawn out the velocity vectors
for both vehicles relative to the Earth.
Now let's add the velocity of the minivan
with respect to the Prius.
And now you should be able to see that there is
an equation that relates these three vectors.
Bo, could you please read that equation?
Bo: The velocity of the minivan with respect to the Earth
is equal to the velocity of the Prius
with respect to the Earth plus the velocity
of the minivan with respect to the Prius.
mr.p: Exactly.
You can see from the vector diagram
that the sum of the velocity of the Prius
with respect to the Earth and the velocity
of the minivan with respect to the Prius
has the same magnitude and direction
as the velocity of the minivan with respect to the Earth.
Billy, could you please solve for
the velocity of the minivan with respect to the Prius?
Billy: We can subtract the velocity of the Prius
with respect to the Earth from both sides,
and we get the velocity of the minivan
with respect to the Prius is equal to
the velocity of the minivan with respect to the Earth
minus the velocity of the Prius with respect to the Earth.
Which is 24 mi/hr East minus 13 mi/hr East
and that is 11 mi/hr East,
which is exactly what I said before.
mr.p: Yes, but now we've drawn the vectors,
we've solved the equation, and we can do the following.
Bobby, what does taking the negative of a vector
do to that vector?
Bobby: That reverses the direction of the vector, right?
mr.p: Yes, it rotates the vector 180 degrees.
In terms of relative motion,
what it does is to switch the order of the subscripts.
In other words...
The negative of the velocity of the Prius
with respect to the Earth is equal to
the velocity of the Earth with respect to the Prius,
switching the order of the subscripts.
And negative East is the same thing as West.
Therefore, the velocity of the Earth
with respect to the Prius is 13 mi/hr West.
Therefore when we look at the video
relative to the Prius again,
notice that the Earth appears to be moving
Westward at 13 mi/hr.
Now we can substitute the velocity of the Earth
with respect to the Prius back into our equation
instead of the negative velocity of the Prius
with respect to the Earth.
What I want to point out here is what's happening
on the right hand side of the equation.
Any time you sum two relative motion velocities
and they share a common subscript
either on the inside or the outside,
that subscript cancels out.
For example,
the velocity of the minivan with respect to the Earth
plus the velocity of the Earth with respect to the Prius.
That Earth, which is on the inside, cancels out,
and we end up with the velocity of the minivan
with respect to the Prius.
The same thing happened actually over here at the beginning.
We have the velocity of the Prius with respect to the Earth
plus the velocity of the minivan with respect to the Prius.
That Prius is a common subscript and it's on the outside.
There it was on the inside, here it's on the outside.
And it now drops out.
We end up with the velocity of the minivan
with respect to the Earth.
And that is what I want you to remember
for more complicated vector addition problems, please.
Now what I'd like to do is solve for
the velocity of the Prius with respect to the minivan.
Again, we are solving for the velocity of the Prius
with respect to the minivan.
Billy, could you please do that?
Billy: Well, isn't the velocity of the Prius
with respect to the minivan just the negative
of the velocity of the minivan with respect to the Prius?
And we already know that,
so it should work out to be -11 mi/hr East,
which is the same thing as 11 mi/hr West, correct?
mr.p: Very nice.
Now that we have the velocity of the minivan
with respect to the Prius,
we can take the negative of that
to get the velocity of the Prius
with respect to the minivan.
Negative 11 mi/hr East is 11 mi/hr West.
That means when we look at the video
with respect to the minivan,
the Prius appears to be moving west at 11 mi/hr.
From the perspective of the driver of the minivan,
it appears as though the Prius is moving
backwards at 11 mi/hr.
So when we look out the window of the minivan,
the Prius appears to move backwards.
To review, let's take a look at the videos of this event
from the perspectives of the minivan and the Prius
at the same time.
The top video is with respect to the Prius,
and the bottom video is with respect to the minivan.
You can see the velocity of the minivan
with respect to the Prius is 11 mi/hr East,
and the velocity of the Prius with respect to the minivan
is 11 mi/hr West.
They have equal magnitudes and opposite directions.
Let's take the Earth out of the video for a moment.
Because neither of these velocities
are relative to the Earth,
they are actually both relative to the other vehicle.
And so when we take the Earth out,
it's actually easier to see the relative velocities.
Again, thank you Aaron Fown of FirstUAV
for volunteering both your time and your quadcopter drone,
which made visualizing these relative motions much easier.
Thank you very much for learning with me today,
I enjoyed learning with you.