Understanding Motion in Two Dimensions
Now you might have been tempted to say I am running like this, rain is falling like that obviously I get wet on the back side. But how do you decide these without knowing the velocities? You might ask how does the velocity matter here? Right? Let’s show you. Let’s say that this mountain that you are running on, you are running with velocity such that your components are 6i + j. So some speed like that which has component like this. Components are usually best ways to deal with vectors, you know that. Let’s say the rain is falling at 3i – 3j for an angle like this right behind you. So very much similar to the case that we discussed. So what are you trying to find here? The ambulate which rain appears come to you. Which means we are trying to relative velocity of the rain with respect to you which means that you must subtract your velocity from the rain’s velocity. You are trying to find vrm. We are trying to do vr-vm. Now with components here, it going to be triple, right?. Your vr is 3i – 3j. Ya? Because the rains is falling down. And your vm is 6i + j, so vr-vm will look like this. So finally you get an answer you have 3-6 over here which is -3 and you have on the other side -3 -1 , which is -4, so -3i -4j as the relative velocity with respect to you. Now observe something over here, right? The i component is minus, which means you took this is a positive direction. If you got a component negative, it means that to you the rain appears to come from this side and of course the vertical components like at. So finally you have, rain is coming in some angle. Let’s find out what that angle is. But crux is here that as far is you are concerned, the rain is going to hit from front. So we just showed you a case where from the ground it all looks like you are running like this and the rain is hitting you from the back, but as far your concern rain is hitting from front. So if you to wear a bag, it is better to wear a bag behind so the exact angle, what is it over here? -3, – 4 right? As the angle is going to be tan-1 (3/4). It’s a famous angle and I know you can calculate that. So the crux over here is that, we cannot answer from which direction the rain will hit you without also knowing your velocity, right? So how does your velocity play here? The idea is that there are more than one ways to get wet. One way is for the rain to hit you. In our case, rain at 3 as its horizontal components velocity. So if you had any velocity less than 3, the rain hits you and you get wet. But what’s the other way? The other way to get wet is you go and hit the rain. And how can you do that? If your horizontal component is more than that of the rain, which is what we saw. Her the horizontal component is 6. So in one sense, you are hitting the rain. So both these are the ways in which you can get wet. But the interesting question is – what would happen if you are exactly running with the velocity that is equal to the horizontal component of the rain? So the rain was coming at 3 and you were also running at 3. Neither would the rain be hitting from the back nor would you be you getting the rain from the front. That’s very interesting case. Isn’t it? Now we began this journey with one question, right? Should we walk or run in the rain? And at that time it seems to be a trivial question. You run as fast as possible, spend as little time as possible, and you will get less wet. But no we have shown you that running as possible might let you hit more raindrops on the way. So the question is not as trivial as it seems, right? But not you are definitely equipped to answer that very question. The final idea that we told that there are more than one ways to get wet. So take some time out to think what might be the best velocity to run at when the rain is coming at various angles.