Intro to Relative Velocity - Online Tutor, Practice Problems & Exam Prep

Relative motion involves calculating the velocity of one object concerning another, using simple addition or subtraction of velocities. The relative velocity equation, $v\_ac=v\_a\_b+v\_b\_c$, helps in solving these problems. Understanding the frame of reference is crucial, as it determines how velocities are measured, leading to insights into concepts like acceleration and momentum in various contexts.

Intro to Relative Motion (Relative Velocity)

Hey, guys. So you may come across some problems in which you have to calculate the velocity of one object that moves relative to or with respect to something else. So I want to introduce you in this video to the idea of relative motion or relative velocity. And really what we're going to see is that it kind of just comes down to simple addition and subtraction of velocities. So let's check it out. What's this whole idea about? Well, whenever we measure velocity, we are measuring it relative to some reference points, which we call a frame of reference. So for example, you're standing on the side of a road, you have a fancy speed gun or a radar gun that cops use to measure how fast you're driving and you're measuring the, you know, the speed of cars passing by. So your reference frame, your frame of reference is the earth. That's the thing that you're measuring those velocities with respect to. And so, in fact, most of the time in problems, your frame of reference is going to be the ground or the earth unless they otherwise tell you. So let's just jump right to an example so I can show you how this works. So imagine that you're an observer, right, and you're standing on the outside of a moving platform or a moving walkway like the kind that you see in airports. You have one of these fancy speed guns with you and you're going to measure the velocities of 3 different people on this moving walkway. So person a is standing still, person b is walking to the right, and person c is walking to the left. So what would our speed gun measure their velocities to be?

Well, the whole idea here is that if you look at \( t \) equals 0, all of these people are kind of lined up along the same position. But then one second later, the walkway has kind of moved everybody to the right. But because some of them are walking forwards, some of them are standing still and some of them are moving backwards, then they're going to be at different distances or different positions one second later. So what happens with person a? Well, what happens is the walkway is just going to move them some distance over, you know, some amount of time in one second. And so because person a is just standing still on the moving walkway that is moving them along at 3 meters per second, that if we pointed our speed gun at person a, it would also show them moving at 3 meters per second. So here's why. Here's a way you can visualize this. So imagine that my pen here is you, the observer. My hand is the moving walkway and person a is my red pen. So as the moving walkway basically moves this person along at 3 meters per second, they are together. And because they are together, they have the same velocity. In fact, whenever you have a stationary or not moving object that is inside or within or on top of another moving object, then they basically have the same velocity. They share the same velocity. And so in other words, another way we can say that is that the moving walkway and person a have the same relative velocity to all other reference frames. So to you, the observer, person a and the moving walkway have the same velocity relative to any other reference frame. And another way of also putting that is that they have the same velocity, then their velocities relative to each other are equal to 0.

So let's move on to person B now. So person B is moving along at 2 meters per second. So what happens with them? Well, in the same amount of time, the same one second, person B is because they're walking forwards, are going to travel a higher distance or greater distance in the same amount of time. Now we know that the velocity is equal to the change in position or the displacement over time. So if they're traveling a farther distance in the same amount of time, then that means that their velocity must be higher. So if we put our speed gun at B, what would we measure? Well, because B is moving at 2 meters per second, because they're walking forwards at 2 meters per second, then their 2 meters per second, in addition to the 3 meters per second, then the walkway is moving them along. Then when we point our speed gun at them, they're not going to be moving at 3, but they're going to be moving at 3 plus 2, which is 5 meters per second. And so what now happens with person c? Well, person c is moving 2 meters per second in the opposite direction. So in the same amount of time, they're going to cover a less distance than the other 2 persons a and b. And so their velocity must be lower than 5. So the idea is basically just the reverse. Their velocity relative to the moving walkway picks up a negative sign. And now what happens is if you were to point your speed gun at person c, it wouldn't measure 5, it wouldn't measure 3, it would measure 3 minus 2 because the velocity is pointed in opposite directions, so they kind of cancel each other out and you would see them moving at 1 meter per second. To summarize everything, really relative velocity is just the addition or subtraction of velocities, and it all depends on which directions those relative velocities are in. That's it for this one, guys. Let me know if you have any questions.

Solving Relative Velocity Problems in 1D

Hey, guys. So in the last video, we were introduced to the idea of relative motion or relative velocity. And what we saw is that to solve relative velocity problems, you're really just adding or subtracting velocities in different reference frames. Now you're going to need to know exactly how to solve relative velocity problems in one dimension. So I'm going to show you how to do that in this video and it really just comes down to setting up what's called the relative velocity equation. Now setting up this equation is the hardest part of solving these kinds of problems because it's tough to figure out exactly which variable you're solving for. But I'm going to show you, there's really just a couple of rules and steps that we can follow, so we can always set up the equation the right way and get the right answer. Let's check it out. So this relative velocity equation is written in the following formats. So the velocity of a relative to c which we write in the notation v_ac is equal to the velocity of a relative to b plus the velocity of b relative to c. Now different books and professors are going to write this using different letters. They might even have it, you know, with some negative signs in a slightly different order. But this is the order that we're going to use in our videos because we think it's the easiest for you to set up and understand. So as I mentioned before, we might actually see different letters and it really kind of just depends on the problem. So for example, in the example that we're going to do down here, we're going to be talking about a car and a truck. You might use the letters c and t instead of a and b. So instead of memorizing the letters, what I want you to do is memorize the rules because those are much more important. And the rules are actually fairly straightforward. So your inner subscripts of the terms on the right side of the equation, so these are the subscripts that are the closest to each other are going to be the same. So notice how I have b and b. And then the outer subscripts on the right side, meaning the subscripts that are farthest away from each other on the right side of the equation are going to equal the subscripts of the term on the left side. So a and c as we have a and c on the left side as well. So really these are just the two rules. If you set this equation up using these two rules, you're always going to get the right answer. So let's see how this works using this example down here. So we're in a car moving at 45 meters per second East relative to the ground, and then we've got a truck that's ahead of you also moving at 60 meters per second. And what's the velocity of the truck relative to your car? So the first step that you're going to do in solving all these problems is you're going to draw a diagram and you're going to identify all of the objects and references. So for example, we've got the road over here like this. We're driving, so you're driving in your car. So we've got one object here, that's the car, and the velocity of your car is equal to v_car = 45 m/s. Now we also have the truck that's ahead of you, and it's also traveling to the east, v_truck = 60 m/s. But there's a third thing here because remember that both of these velocities ground over here and we've also got the truck. So that's the first step. So now we identify all the objects in the references. The next we have to do is write each of the given velocities with the notation. Remember that the subscript notation means you're always going to measure or you're always going to write the velocity of one thing relative to another. So when it says the velocity of the car, what it's actually saying is the velocity of the car relative to the ground. So this isn't just v_c, it's v_cg, and in the similar way, this is v_tg. So what do we ask for in the problem? So we're asked for the velocity of the truck relative to your car. So if we use the same notation, it's going to be v_tc measured by or relative to c. That's actually what we're asked for in the problem. So v_tc is going to be our target variable, and we know what v_cg is, that's just 45. And we know v_tg is equal to 60. So now that brings us to the 3rd step, which is we have to write the relative velocity equation according to the rules for inner and outer subscripts over here. So we're looking for v_tc, so we're going to write that on the left side of the equation. So v_tc is going to be equal to, and now what we'd have to do is we're going to have to set up the terms so that the outer subscripts, basically the ones that are farthest away from each other on the right hand side, are the same as the subscripts on the left side. So what I need is I need a term that has v as the first or t as the first subscript, and then I need the other term to have c as the last subscript, right? So if you look through my terms here, I've got one term that has a T in the front and that's v_tg. Right? So if we look through our variables here, this v_tc is what I'm looking for. This v_tg is the 60 meters per second and I need v_gc. So I've already got these. I've already got this variable covered here. Now if we notice, this variable that we're given, the 45, isn't v_gc. It's v_cg. It's the same letters, but they're just flipped backward. So how do we deal with this? Well, if you're ever given a velocity with the correct subscripts, but just in the opposite order, then you can flip or reverse the subscripts. And whenever you do that, whenever you flip the subscripts, then you're basically just flipping the sign of the number from negative to positive or vice versa. So in general, v_ab is gonna be the negative of v_ba. So what that means in our example here is that if we want v_gc and we're given v_cg, then we can always just reverse the order of the subscript. So v_gc is just gonna be negative 45. And now that we have this variable and this number, now we can actually plug it into our equation and solve. So our v_tc is just going to be the 60, the truck relative to the ground, plus negative 45. And so what happens is v_tc is equal to 15 meters per second. So that's the answer. That's the velocity of the truck relative to you. So let's summarize and actually put yourself in this scenario here. So you're driving along in your car at 45 meters per second. That's relative to the ground, which means if somebody were on the ground with their speed gun measuring your velocity, it would read 45 meters per second. But if you had your own speed gun and you were measuring the truck ahead of you, it wouldn't read 60, it would read 15 meters per second faster than you. That's really what that 15 meters per second means. Alright guys, that's it for this one. Let me know if you have any questions.

A boat on a river is traveling from a pier to a point 500 m upstream (against the river's current). The current flows at 4 m/s. If the boat makes the trip in 250 s, what is the speed of the boat relative to the water?

• A. 2 m/s
• B. 0 m/s
• C. 6 m/s
• D. 12 m/s

Flying in the Wind

Hey, guys. Let's take a look at this problem here. So we got these 2 cities. Without any wind, this airliner can make the trip between them in some amount of time, then a wind starts blowing. So this 225 kilometer wind that blows from west to east. We're trying to figure out how long the trip from a to b is going to take now that the wind is blowing. So we're going to solve this because we have some relative motions, some relative to the air, stuff like that, using our relative velocity steps. But there is a little bit of a before and after case that's going on here. So we're going to draw these diagrams, but I'm actually going to draw 2, one for the before situation and one for the after. So let's get to it. The before situation that happens is in this first part of the problem where we're told that without any wind, the trip is, you know, the airliner can make the trip in some amount of time. So the before situation is with no wind. And then the after is going to be when the wind starts blowing. So this is going to be some from the wind. So let's check this out here. So we've got these two cities. Right? So we've got a and b. We know b lies directly west or east of a. And so the delta x, we're told, is equal to 2775 kilometers. And we know the time that it takes to travel between these two cities is 3.3 hours. So what happens is if the plane is just traveling through the air like this, this is the plane like this, then we can actually just straight up figure out what the velocity is just by doing delta x over delta t. So it's just 2775 over 3.3 hours. And so the velocity of the plane through the air is 841 kilometers per hour. So that's the speed of the plane, right, without any wind. But now what happens is the wind is going to start blowing. And so now there's going to be a couple of velocities to consider. There's going to be some relative velocities. Let's draw the same exact diagram. So we've got, again, city a and city b and we know the delta x. We know that this displacement is 2775. What's different now is because there's some wind that's potentially helping or changing the velocity of the plane, this delta t is going to be different. So that's what we're actually asked for in the problem. How long does this trip now take with some wind blowing? So I'm going to call this delta t initial. That's the 3.3 hours. But what I'm really looking for is delta t final. What is the change? What is the amount of time that it takes now that the wind is blowing? So this is really what I'm interested in. Okay. So what's going on here? So now we're just going to go ahead and stick to the steps for solving relative motion problems. So we've got this diagram here. Now we just need to identify the objects and the references. So we've got the plane that is traveling. So we've got this plane over here. That's one of our objects. We're told that the plane in the second part of the problem has the same speed relative to the air, which means that the air is actually itself a reference or an object. So we're going to call the air the object here. And we're told that the air also has a speed. The velocity of that wind that's blowing from west to east is 225. And so it picks you know, it's positive because it points in the same direction. Right? So the velocity of the air. So the next question is what is the velocity of that air measured relative to? Well, so our third reference is going to be the ground. So that's the third reference here, the ground. So we got the plane traveling through the air, but the air is traveling relative to the ground. So those are our 3 objects and references. So now we're going to write each of their velocities with the given subscript notations. So we're told that the velocity of the plane relative to the air is going to be the same as before. So that means that this variable over here was V_PA, and so that's 841. And the air is measured relative to the ground. That's the, you know, the wind speed or whatever. And so that's V_AG. So we've got our two variables here. So we've got V_PA, which we know is 841. We've got V_AG, which we know is 225. So remember, we're going to need a third one to start writing a relative velocity equation. So what's that third variable? Well, the only other thing that we can use is the fact that it covers some distance which is on the ground. Basically the ground distance between a and b in some amount of time. So what this means is that the velocity of the plane through the air and the velocity of the air relative to the ground can combine, and they'll produce a velocity of the plane relative to the ground. So that's V_PG. So both of these things here combine to form V_PG. That's the third variable. So this is actually what we're looking for here, our V_PG. Well, at least we're not given it. So think about this, right? If we're trying to figure out the total amount of time that it takes for the airplane to travel this distance here, then we're going to need to know its plane to ground speed. So if I can figure out what this V_PG is by using my relative velocity equations, then I can actually just go ahead and solve for delta t. So I know this is going to be delta x over t final. So therefore, my delta t final is just going to be delta x over the velocity of the plane relative to the ground. So again, if I can figure out this V_PG, then I can actually go ahead and and figure out what this time is. So now that brings us to the third step, which is we're going to set up the relative velocity equation. So V_PG, if this is what we're looking for, and now we're just going to stick this to our rules. We want the outer subscripts to be p and g. That means I want the first subscript of the first term on the right to be p, And then I want the last subscript to be g. And then the letter that goes inside of the inside subscripts is just going to be the only letter that we haven't used yet, which is a. So this is going to be a and a. So now the inner subscripts are the same, the outer ones are the same as the first term. So now we're just going to go through and figure out if we have all of our terms, if we need to flip anything, stuff like that. So V_PG, that's what I'm looking for. This is going to be V_PA. This is the 841 plus V_AG. So I actually have all of those numbers here. So I can just go ahead and straight up plug them in. So V_PG is just equal to 841 plus the 225. And if you work this out, you get 1066. So if you think about this, the velocity of the plane relative to the ground is just the velocity of the plane relative to the air, plus the velocity that the wind is basically helping it move along the ground. So these things kind of add together to form a faster velocity. Right? So it's kind of like if you were running and, you know, you have wind at your back, it kind of helps you and pushes you along and, you know, go a little faster. So now that we have this velocity here, now I can just plug it back into our equation and solve for delta t. So our delta t final is going to be the same distance, 2775 between the two cities divided now by 1066. And if you work this out, you're going to get 2.6 hours. So that's how long it takes. That makes some sense. If the wind is helping you along, then whereas before it took you 3.3 hours to travel the distance between them, now it only takes 2.6 because the wind is kind of helping you push you a little bit faster. So our answer choice is b, and that's it for this one, guys. Let me know if you have any questions.

Here’s what students ask on this topic:

What is relative velocity and how is it calculated?

Relative velocity is the velocity of one object as observed from another moving object. It is calculated by adding or subtracting the velocities of the objects involved, depending on their directions. The general formula for relative velocity is given by:

$v_{\mathrm{ac}}=v_{\mathrm{ab}}+v_{\mathrm{bc}}$

Here, $v_{\mathrm{ac}}$ is the velocity of object A relative to object C, $v_{\mathrm{ab}}$ is the velocity of object A relative to object B, and $v_{\mathrm{bc}}$ is the velocity of object B relative to object C. Understanding the frame of reference is crucial for accurate calculations.

How do you solve relative velocity problems in one dimension?

To solve relative velocity problems in one dimension, follow these steps:

1. Draw a diagram and identify all objects and reference frames.
2. Write each given velocity using subscript notation, indicating the object and its reference frame (e.g., $v_{\mathrm{cg}}$ for the velocity of the car relative to the ground).
3. Set up the relative velocity equation using the rules for inner and outer subscripts:

$v_{\mathrm{tc}}=v_{\mathrm{tg}}+v_{\mathrm{gc}}$

Here, $v_{\mathrm{tc}}$ is the velocity of the truck relative to the car, $v_{\mathrm{tg}}$ is the velocity of the truck relative to the ground, and $v_{\mathrm{gc}}$ is the velocity of the ground relative to the car.

If needed, reverse the subscripts and change the sign of the velocity. Finally, solve the equation to find the desired relative velocity.

What is a frame of reference in relative velocity problems?

A frame of reference is a coordinate system or viewpoint from which the position and motion of objects are measured. In relative velocity problems, the frame of reference determines how velocities are observed and calculated. For example, if you are standing on the ground and measuring the speed of a car, the ground is your frame of reference. Most problems use the ground or Earth as the default frame of reference unless stated otherwise. Understanding the frame of reference is crucial for accurately determining relative velocities and solving related problems.

How do you handle opposite directions in relative velocity calculations?

When dealing with opposite directions in relative velocity calculations, you need to assign a negative sign to the velocity of the object moving in the opposite direction. For example, if one object is moving at 3 m/s to the right and another at 2 m/s to the left, their relative velocity is calculated by subtracting the leftward velocity from the rightward velocity:

$3-2=1$ m/s.

This approach ensures that the directions are correctly accounted for, leading to accurate relative velocity calculations.

What is the significance of the relative velocity equation?

The relative velocity equation is significant because it provides a systematic way to calculate the velocity of one object relative to another. The equation:

$v_{\mathrm{ac}}=v_{\mathrm{ab}}+v_{\mathrm{bc}}$

allows you to break down complex motion into simpler components, making it easier to solve problems involving multiple moving objects. By understanding and applying this equation, you can accurately determine how objects move relative to each other, which is essential for analyzing motion in various contexts, such as transportation, sports, and physics experiments.