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.