Rain man problem

suppose we have a situation where it's raining and the rainfall speed is 5 m/s and suppose you are traveling in your bike 12 m/s towards the right the question is in what direction should you hold your umbrella at first this sounds like a random and an absurd question I mean it's reasonable to ask why can't we just hold our umbrella this way just like normal right well if you hold your amulet this way when you're traveling in a bike you end up getting wet it's easier to see that if we could just isolate one raindrop let's do that it's isolate one raindrop now this drop is falling down like this now if you were still on the ground not moving then that raindrop wouldn't have bothered you however remember you are moving towards the right so if you wait for some time not only will the drop fall down but in that time you and your bike would have also gone forward to hit that raindrop in other words your front part of the bike could get wet because of this and so that is not the direction in which you should hold your umbrella if you don't want to get wet so let's put that umbrella down and let's think about this the key mode to solving this problem would be looking at things from the bikers perspective all right in other words we have to look at the direction of the rain fall as seen from the bike or with respect to the bike who do you see that this is really a question in disguise about relative motion the real question is what is the direction of the rainfall with respect to the bike or relative to the bike so we need to now jump into the bikes perspective and look at things and we've already done one video on relative motion in two dimensions and if you have not watched that it's better to first watch that and then come back over here alright so let's jump into the bikes point of view this is my poor attempt of showing the bikes perspective this is your arm this is your shoulder its excuse the drawing over here what will you what will it look like from the bikes point of view well you don't see the bike moving instead the rest of the world will go backwards so you will see the ground going back you also see the air going backwards that's why you get the breeze and everything and notice since on the ground you the bike is going forward 12 meters per second with respect to the bike the ground is going backwards 12 meters per second so keep that in mind okay now let's look at what happens to a raindrop let's pick one raindrop over here and also let's put a partition over here okay super we have to calculate what direction the raindrop falls from the bikes perspective to do that we will wait for one second so if we wait for one second you will see that this raindrop would fall down five meters because it's falling down five meters per second so let's just write that down over here so this would be five meters in one second but remember in that one second the ground and the raindrop and the air everything is going backwards 12 meters so the whole thing goes back 12 meters so let's put that as well so the whole thing would go 12 meters back in one second and so now you can see that one second ago the drop was here now it's over here so as seen from the bike the raindrop doesn't fall straight down but instead it goes this way so this means as seen from the bike the rainfall is no longer happening downwards it's happening at an angle like this and this could be a very familiar situation if you ever been in a car or a bus while it was raining and if you looked outside you really see that the raindrops no longer fall vertically down but instead there'll be angle little backwards okay alright so our goal now is to calculate what this angle is going to be so let's do that what is this angle equal let's call this angle is alpha and we can calculate this using trigonometry this is a right angle triangle we know the opposite side we know the adjacent side which ratio connects opposite side edges inside ooh 10 so we can say tan alpha is equal to the opposite side that's 12 / the adjacent side divided by five so that gives us two point four and so alpha will be equal to tan inverse of two point four and if you look up a trigonometry table this angle turns out to be roughly sixty seven point four degrees and so now we know exactly at what angle the rainfall is happening from the bikes perspective and so from the bikes perspective the rainfall might look somewhat like this so what this way this is what it looks like and it's for that reason if you have to hold your umbrella it should be pointed a little bit forward and since we know this angle over here the rainfall is making an angle of alpha sixty seven point four degrees with respect to the vertical you must also hold your umbrella at that same angle forward so you should tilt your umbrella at the same angle alpha forward and there's our solution now one small note we could have arrived at this at this result directly by using the general expression that we derived in a previous video and that was the relative velocity of the rain with respect to the bike that is equal to the velocity of the rim as seen from the ground minus the velocity of the bike as seen from the ground and if you had used this expression we would get the same result let's see we are adding VR so over here there's we are and the negative of VB well VB is 12 meter per second forwards so negative of that would be 12 meter per second backwards C so when you add this you get this vector so we get the same result another thing we could understand over here is you could calculate this length if you use Pythagoras theorem then this is the hypotenuse the hypotenuse would be equal to things are getting crowded the hypotenuse is equal to 12 squared plus v square Pythagoras that's going to be 13 so this this length would be 13 which means that the rain would travel 30 meters this way in one second which means from your perspective the rainfall is happening at 30 meters per second it's more than double the velocity as seen from the ground and so this could really hurt you so traveling very fast in the rainfall may not be a great idea