in this article we're going to talk about relative velocity so when you hear the word relative velocity what do you think of what I think of you're talking about the velocity of an object relative to something else and whenever you want to calculate the relative velocity is basically the difference between two velocities so let's say if we have the variable vba you need to understand what this means what this means is you're talking about the velocity of object b with respect to object a so b is the object that you're considering that's the object in focus a is the reference frame or the frame of reference so how can we calculate v ba how can we find it so this is the velocity of b with respect to a and it's basically the difference between v b and v a now if you see me write v b throughout the remainder of this video what I mean is v b is really the velocity of b with respect to the earth so let's say if a train is moving at 30 miles per hour with respect to earth if you're standing on the ground not moving to you the train is going to appear to be moving at 30 miles per hour so if I just write one letter one subscript then assume that it's always with respect to something that's stationary in this case a good example will be the earth so v a is the velocity of object a with respect to the earth now let's say if we want to define vca that's the velocity of c with respect to a so you can calculate it by subtracting vc and va now if you want to find v of afc that's the velocity of object a with respect to c where c is the frame of reference then this is going to be v a minus v c so if you want to find the velocity of object c with respect to b that's vc minus vb and the velocity of object b with respect to c is vb minus vc so basically relative velocity is just the difference between the velocity of the object minus the velocity of the frame of reference which i'm just going to put fr so that's the formula to calculate relative velocity and we're going to go over some examples that's going to illustrate so let's say this is a train or train cart and as a person inside it let's call this person person a and the train is moving at 50 miles per hour with respect to the earth and there's another train train b that's moving at 60 miles per hour let's say there's a person on this train as well now if they're both moving in the same direction calculate the relative velocity of b with respect to a and also find the relative velocity of a with respect to b so train b is moving 10 miles faster than train a so the velocity of b with respect to a is simply 10 miles per hour the way you would calculate it is it's going to be vb minus va vb is 50. va i mean va is 50. I take that back and uh vb is 60. so it's going to be 60 minus 50 which is 10 miles per hour so that's the velocity of b with respect to a train b is moving 10 miles per hour faster than a and the velocity of a with respect to b it's va minus vb that's 50 minus 60 so that's negative 10. so a is moving 10 miles per hour slower than b so that's how you could find the velocity of a with respect to b now let's think about what this means so let's say this is a and this is b so a is moving at 50 miles per hour east and b is moving at 60. now we said that v the velocity of b with respect to a is positive 10. so that means that b is going 10 miles per hour faster than a if they're moving in the same direction so in one hour a is going to travel a distance of 50 miles and our reference frame is a so b is going to travel a distance of 60 miles so in one hour b is going to be 10 miles ahead of a so if you're a person on train b and you're looking at someone on train a to you it's going to appear as if you're moving 10 miles faster than a imagine if you're in a car and you're on the highway going at 60 miles per hour and there's another car beside you traveling at 50 miles per hour now eventually you're gonna move past the other driver because you're going faster but you're not going to move past him quickly it's going to happen over time but slowly you're going to move past him and in one hour you're going to be 10 miles ahead of person a so that is your relative velocity with respect to a is 10 miles per hour so every hour you're going to gain 10 miles from person a now let's analyze it if b is the reference frame the velocity of a with respect to b where b is the reference frame is negative 10. so in one hour a is going to travel a distance of 50 miles and in one hour b is going to travel a distance of 60 miles so keep in mind b is the reference frame so with respect to person b someone in train b is going to look at someone in train a and it appears as if train a is moving towards the left with respect to b because before they were at the same location but now if you're someone on train b and you're looking at the person on train a it appears as if with respect to you that the person in train a is moving towards the left and in one hour he's going to travel 10 miles t  owards the left so with respect to u his displacement is negative 10. and thus that's why we can see that it's negative so with respect to someone on train b a person on train a will appear to be moving towards the left and that's why we have a negative relative velocity and the other example where a was the reference frame a person on train a and looking at the person on train b sees that person moving 10 miles per hour away from them towards the right and so that's why the relative velocity of b with respect to a is positive because the person on b is moving towards the right away from person a and so hopefully this helps you to understand the sign of relative velocity and what it means now let's work on another example so let's say if there's a person on train a and they're moving east at 50 miles per hour so that's a va with respect to the earth and then there's another train with a person on it let's call that train b and he's moving 40 miles per hour west so the velocity of b with respect to the earth is negative 40 because he's going towards the left or the negative x direction so here's a question for you what is the relative velocity of a with respect to b and what is the velocity of b with respect to a now before you actually calculate it predict which one is going to be positive and which one is going to be negative now a is moving towards the right with respect to b so as time progresses a is going to keep moving to the right so the velocity of a with respect to b should be a positive answer now b is moving towards the left and over time it's going to eventually be left of person a so vb of with respect to a is going to be negative now to actually calculate v of a with respect to b that's going to be v a minus v b so it's 50 minus negative 40 which is 50 plus 40 so that's 90. so in one hour a is going to move 90 miles closer to b so let's say if they were initially 200 miles apart in one hour a is going to move 50 miles and also b is going to move 40 miles towards the left so now if a is located at this point and b is right here they are now 110 miles away from each other so therefore the distance went from 200 to 110 the distance decreased by 90. so therefore the relative velocity is 90 miles per hour every hour the distance between them will decrease by 90. now the velocity of b with respect to a that's going to be v b minus v a where v b is negative 40 minus v a which is positive 50 so that's negative 90. so if we analyze the velocity of a with respect to b it's positive 90. a person on train a every hour is going to move 90 miles closer towards b in towards the right so in the positive x direction so that's why it's positive 90 miles per hour now looking at the velocity of b with respect to a a person on train b is moving towards train a but to the left so because they're moving towards the left every hour they're going to move 90 miles closer to the left towards train a so that's why the velocity of b with respect to a is negative 90. now let me give another practice problem so let's say if we have another person on a train well let's say the person's on a car or something and this person well let's say the car is moving at 40 miles per hour with respect to the earth and the person driving the car throws a ball east at 15 miles per hour with respect to the car and there's another car approaching the first car and let's say this car is moving 60 miles per hour towards the west so what I want you to do is I want you to find the velocity of the ball with respect to this person so let's say this is car excuse me this is car a this is the ball b and this is car c so we could distinguish everything so what is the velocity the relative velocity of the ball with respect to car c feel free to pause the video to get this uh answer so let's define what we have the 40 miles per hour represents what what variable does it represent that's the velocity of card a with respect to the earth the 60 miles per hour is the speed of car c but if you want the velocity of car c with respect to the earth it's negative 60 because it's moving towards the left now the 15 miles per hour that's the velocity of ball b not with respect to the earth but with respect to car a so our goal is to find the velocity of the ball with respect to car c in order to find the velocity of the ball with respect to the car we need to use this equation vb minus vc now we have vc but we don't have vb so we need to find another way of getting vb so we need to use an equation that relates vba with vb now vba the velocity of the ball with respect to car a is the velocity of the ball with respect to the earth minus the velocity of car a with respect to the earth so vba we have positive 15 it's going towards the right vb we don't know but va is 40. so in order to find vb we got to add 40 to both sides so therefore vb is 15 plus 40 so it's 55 miles per hour and it makes sense because if the car is already moving at 40 miles per hour and you throw