### Video Transcript

In this video, we’re going to learn
about relative motion, what it is, how to calculate it, and how to keep track of
direction to help us solve relative motion exercises.

To start out, imagine that your dog
Fido has a bad habit of running away. After many hours spent searching
for and finding Fido, you see a product which may offer a solution. The product, called Never Run Away,
is made for dog owners in just your situation. It consists of a battery-powered
bracelet for you and a battery-powered collar for your dog. The system is set up to require
that the relative velocity between the bracelet that you wear and the collar that
Fido is wearing is less than or equal to 20 kilometers per hour. If that velocity is ever exceeded,
the bracelet and the collar both emit wireless signals to the local authorities
announcing that there’s been a lost dog.

One day at the park, while you and
Fido are using the Never Run Away system, Fido sees a river and jumps in, begins to
float downstream at five kilometers per hour. Thankful for a bit of a break from
watching Fido, you decide to hop on your bicycle and take off riding perpendicularly
to the river. You’d like to know how fast you can
ride on your bicycle without exceeding the maximum relative velocity between you and
Fido and thereby sending out a lost dog alert.

To figure out what that speed is,
we’ll need to understand relative motion. Relative motion is the movement of
an object with respect to some other object or reference point that is also in
motion. Examples could include a train
moving to the east relative to a car driving south, or salmon swimming upriver
relative to the flow of the river’s current, or we could find the motion of an
airplane relative to a person walking on the ground. In relative motion examples, we’ll
find more than one object that’s in motion. As we look into relative motion,
we’ll find that, in general, it has to do with frames of reference. Here’s what that means.

Say that you go out for a walk, and
you move ahead at five kilometers per hour. Let’s consider what this number
five kilometers per hour means. Because in order to get this
number, what we’ve done, perhaps without thinking much about it, is we’ve set up a
frame of reference fixed somewhere on the Earth’s surface. This number that we record for our
speed is in reference to that reference frame. It’s perfectly normal for us in
everyday life to specify speeds without consciously thinking of our frame of
reference. But the point is we need a frame of
reference in order to be able to have a speed relative to it.

To show how important this notion
of reference frame is when it comes the relative motion, imagine that we move the
reference frame from a fixed location on Earth’s surface onto ourselves as we
walk. If the reference frame we use to
measure our speed moves with us, then what does our speed become? At that point, our speed is zero
because the reference frame moves with us. The motion we’ve talked about so
far has just been a motion of a single object, us as we walk. Let’s imagine we add in the motion
of a friend running the opposite direction as we walk at twice the speed. When it comes to calculating the
relative motion between us and our friend, there are two ways we can go about
it.

One way is to use a reference frame
fixed to the Earth for all motion involved. In that case, we compare our motion
with the motion of our friend on the same set of coordinate axes to solve for the
relative or overall motion involved. That’s method number one for
solving relative motion exercises. A second approach we can use is to
fix the coordinate frame onto one of the moving objects themselves. Say we put our reference frame on
your friend as he or she moves. In that case, their speed is no
longer 10 kilometers per hour and our speed is no longer five kilometers per
hour. Rather the friend’s speed goes to
zero. And our speed relative to that
moving frame goes to 15 kilometers an hour.

When we work on relative motion
exercises, we have the freedom to choose where we want to establish our frame of
reference. We do need to establish one
somewhere, but where exactly we do that is up to us. Let’s work through an example of
relative motion to get practice with these ideas.

Raindrops fall vertically at 4.5
meters per second relative to the Earth. An observer in a car moves through
the rain at 17.9 meters per second in a straight line. The observer measures the velocity
of the raindrops relative to the car. What speed does the observer
measure the raindrops to move at? At what angle below the horizontal
does the observer measure as the direction of the raindrops’ velocity?

In part one of this exercise, we
want to solve for the observed speed of the raindrops. And then, we want to solve for the
angle below the horizontal at which they seem to be moving to the observer in the
car. We can call the speed 𝑣 and this
angle 𝜃. Let’s start by drawing a diagram of
this situation. In this exercise, we’re riding in a
car driving in a straight line at a speed we can call 𝑣 sub 𝑐 of 17.9 meters per
second. As we drive along, raindrops fall
onto the car, falling with a speed we can call 𝑣 sub 𝑑 of 4.5 meters per second
vertically. We want to solve for the relative
motion between ourselves in the moving car and the falling raindrops.

What we can do to start is to take
these two velocity vectors, the velocity of our car to the right and the velocity of
the raindrops falling down, and put them tip to tail. Here we have it: 𝑣 sub 𝑐 leading
up to 𝑣 sub 𝑑. And we see that if we were to add
these two vectors together, graphically they would result in a vector like this. If we were to call this resultant
vector 𝑅, then it’s the magnitude of this vector that equals 𝑣, the speed of the
raindrops relative to us in the car that we want to solve for. Since the two directions of our
relative motion are perpendicular to one another, we can solve for the magnitude
we’re interested in by referencing the Pythagorean theorem. That given a right triangle with
sides 𝐴 and 𝐵 and a hypotenuse 𝐶, the length of 𝐶 squared is equal to 𝐴 squared
plus 𝐵 squared.

In our case, 𝑣 squared is equal to
call 𝑣 sub 𝑐 squared plus 𝑣 sub 𝑑 squared. And if we take the square root of
both sides of this equation, we have an expression for 𝑣 in terms of known
quantities 𝑣 sub 𝑐 and 𝑣 sub 𝑑. When we plug in the values given
for 𝑣 sub 𝑐 and 𝑣 sub 𝑑 and enter this expression on our calculator, we find
that, to two significant figures, 𝑣 is equal 18 meters per second. That’s the speed of the raindrops
relative to us as we observe them falling from our moving car.

Next, we want to calculate the
angle 𝜃 that the raindrops make relative to the horizontal as we observe them
hitting our windshield. To figure that out, let’s consider
what it would look like to see those raindrops hit our windshield. If we arrange our velocity vectors
slightly differently so now they’re tip to tip, ending at the same point, we can see
that as the raindrops strike our windshield, they would look as though they’re
moving down and towards us. And it’s the angle 𝜃 between a
dotted horizontal line and that direction of the rain that we want to solve for.

Looking at this diagram, notice
that 𝜃 is also equal to the angle between 𝑣 sub 𝑐 and the rain direction as we
would observe it. Looking at this right triangle, we
could write that the tangent of the angle 𝜃 is equal to 𝑣 sub 𝑑, the opposite
side, divided by 𝑣 sub 𝑐, the adjacent side. If we take the inverse tangent of
both sides of this equation, that term cancels with the tangent function in the left
side of our equation, leaving us with an expression for 𝜃. That it’s equal to the inverse
tangent of 𝑣 sub 𝑑 over 𝑣 sub 𝑐. When we plug in the known values
for 𝑣 sub 𝑑 and 𝑣 sub 𝑐 and calculate this angle, we find that, to two
significant figures, it’s 14 degrees. That’s the angle below the
horizontal at which we observe the oncoming raindrops.

Let’s take a moment now to
summarize what we’ve learnt about relative motion.

We’ve seen that relative motion is
the movement of an object with respect to something else that’s also moving. We’ve also seen that we can
calculate relative motion in two different ways. In the first case, when we have two
objects in relative motion to one another, we can set our coordinate axes outside
either those objects at some fixed point. We measure the motion of the two
objects relative to that fixed frame and use that method to calculate their relative
motion. The second method is to attach our
coordinate frame to one of the objects in motion. In that case, that object
effectively is still. And the motion of the second object
compared to it is the relative motion between those two. Both of these methods work and
which one is the best to choose often comes down to the way our scenario is
framed. And when we talk about relative
motion, signs — positive or negative, north, south, east, or west compass directions
— are often important. So be sure to keep direction in
mind as we work through relative motion examples.