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Relative velocity in 2D

How fast is the sheep moving as seen by the lion? In previous videos, we have already seen how to calculate relative velocity in 1D. In this video let's extend this concept to the next level, to 2D motion. Created by Mahesh Shenoy.

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Video transcript

if you have things moving in the same direction like an angry sheep chasing a lion or maybe things moving in exactly opposite direction like these guys approaching each other in such cases we can calculate the relative velocity between them just by subtracting them with signs and we have covered this in earlier videos in 1 dimensional motion and if you've not seen them or you need a refresher then please go back and watch those videos and then come back over here because in this videos we gonna talk about relative velocities between things that are moving at some angle with respect to each other so let's take an example so imagine we have a situation where a sheep and a lion are moving perpendicular to each other on streets yeah they're animals running on the street what we would be interested now is in calculating the relative velocities in this example like for example what is the velocity of of the sheep as seen by the lion from the Lions perspective what would that be and how do we even calculate that could we just like directly subtract can you do that let's see what we can do is we can use the same concept that we did before the concept was this if you want to calculate from velocity of the sheep from lions perspective then we have to look at things from Lions perspective the first thing is the lion does not see itself moving as far as the line zipline is concerned it's at rest and this is familiar I mean you're jogging you don't see yourself moving right instead you would see the rest of the world going backwards similarly over here lion would consider itself to be at rest and the whole world including this roar would be going backwards at 3 meters per second all right so let's just look at what that looks like so here it is let's jump into the Lions reference frame or Lions perspective and from the Lions perspective you would see you would see the whole road going backwards this way and if you have if you've used GPS then you see something very similar right you don't see yourself moving you see the rest of the world moving now comes the big question what will the lion see the Sheep doing I mean think about the ship is moving 4 meters per second on the road on the ground but the ground itself is traveling backwards three meters per second so to figure out the relative velocity of the ship seen from the lines point of view we will wait for one second and figure out where she ends up all right let's do that so in one second we would see that the ship would have traveled four meters on the ground all right so let's just write that down on the ground the ship would have traveled four meters here it is four meters on the ground per second but in that one second you will also see that the road and the ship would have gone downwards downwards three meters okay so let's put that in as well so the road and the whole ground oops whole ground goes three meters in one second so now here comes a big question how much did the ship move well notice the ship was initially over here but now it's over here so if you look from the Lions point of view the ship has moved in this direction this way so this means that the ship is no longer traveling towards the right instead lion would see the Sheep moving this way this direction that is the direction of the velocity of the Sheep and at first it might sound a little little weird as do you know why would the Sheep be doing that but let's play an animation to really convince you that that's really what's happening so here it is here's the perspective of Lion all right this is not very convincing isn't it okay let's look at one more time this time we will see some ghost images ah you can clearly see now that the Sheep is indeed moving at an angle isn't it so watch it over and over again until you really get convinced about this all right okay now the next thing would be to calculate this how much is this distance well this is a right angle triangle and in right angle triangle we can use the Pythagorean theorem this is our hypotenuse so we would say the hypotenuse squared the Pythagoras theorem says the hypotenuse squared is equal to the sum of the squares of the other two sides so that should be equal to the square 4 squared plus 3 squared and that gives us 16 plus 9 which is 25 or the hypotenuse turns out to be the square root of 25 which is just 5 which means this length is 5 meters and this happened in one second right so we could say it's 5 meters per second this means that the lion sees the sheep traveling 5 meters this way in one second in other words the relative velocity of the sheep as seen from the Lions perspective is 5 meters per second this way all right now since we're talking about velocity it's not just the magnitude we also want to calculate the direction I mean know we understand the direction is this way but what exactly is this direction so we can answer that by calculating the angle so let's calculate this angle for example how much is this angle so we can use trigonometry we can use sine cause we usually prefer to use tan it's not a necessity you can use anything you want but if I use tan we will get the opposite side 3 oops let's put the proper colors 3/4 that makes theta equals tan inverse of 3 sorry 3/4 and there we have it so if someone asks you now what's the relative velocity of the sheep as seen by the lion we could just say it's 5 meters per second at an angle of tan inverse of 3 by 4 with the horizontal or something like that all right now let's put the whole thing this whole result in terms of symbols okay remember how I used to write symbols for relative velocity well we are calculating velocity of the sheep with respect to right so we would write it as velocity of the sheep sheep with respect to the lion the only thing we have to do now is make sure that we put vector signs because we are dealing in looking at things in two dimensions right so we have to use vectors and if you're not familiar with vector vector so if you're uncomfortable with vectors then it would be better if you could first watch those videos on vectors and then come back over here so anyways let's try to write this now what is this equal to this vector is this vector which we have drawn five meter per second so how did we get this is the question well we added this vector and this vector right this is a triangle law vector addition so we have to add the white vector but white vector is just the velocity of the sheep as seen from the ground so that is equal to velocity of the sheep as seen from the ground plus because we are adding this vector what is this green one oh that's the negative of this vector right I mean since the wine is going upwards three meters per second that's the reason the ground goes backwards three meters per second from the Lions point of view right so this would be the negative of the velocity of the lion as seen from the ground and there we have it this is a general result just to make it more concise we have a plus of a - we'll rewrite this read this as velocity of sheep with respect to the lion as velocity of the sheep - velocity of the lion all right and this is the same as this okay okay and the reason I want to box this expression is because this is very similar to what we got in one dimension do you remember in one dimension we got we a be equal to VA minus VB well guess what we got the exact same result we of SL equals vs minus we L the only difference here is that we have to subtract vectorially using the triangle law or the parallelogram law and in fact even in 1d we do the same thing the only difference is over there we just use signs to take care of directions that's it but it's exactly the same thing and so this is how in general you can figure out relative velocity between things even when they're not moving in the same direction