Falling Balls in Disguise
Let’s begin with a question. Let’s say that you have two identical balls. Say something like these. And now, I say that I am going to drop one of these, but then I am going to throw the other horizontally and catch it. Then, the question is if I let both happen and reach the ground from the same height, which will reach first? Because you probably already know that if I just drop the two, then they will reach at the same time even if they have been of different masses. So, in other words, them being identical is not so important here, right? But now for the real question which is – if I do drop one and throw the other horizontally, which will reach first? Now you might have been tempted to say that the ball that is falling down vertically will reach the ground first, because you will argue that that ball is covering a shorter distance than this ball, or you might have said that they both reach at the same time simply because statistically most questions like this have that as the answer. But if you did do that, would you be able to give yourself a convincing argument as to why that is true. So let’s get to the bottom of this. So let’s take our axis over here and let’s assume we drop one of those balls right on the y-axis and the other one is given a horizontal velocity so that it goes and falls somewhere over there. Now the crux is something that we have been telling you again, and again, and again. Which is, the horizontal and the vertical motions are independent. If you just consider the vertical motion, what’s the initial velocity of both the balls? It’s zero right? Both the balls have an initial vertical velocity of zero. And what’s their acceleration? Both of them have the same acceleration of 9.8 m/s2 downwards. Then could they reach the x-axis at different times? No, right? But you might ask – doesn’t the horizontal velocity matter itself? But it does matter. All that the horizontal velocity of the ball determines is how far along the x-axis the ball will be when it does reach the x-axis. Now to visualize this better, let’s see it this way. Right? Let’s have one more ball which is rolling on the x-axis with the same horizontal velocity as the ball on the top, which was thrown. Now the point to notice here is that both these balls have no change in their horizontal velocity and why is that? Because there is no acceleration in that direction. All the acceleration is downwards, so it’s going to have no impact on the horizontal direction. So, pretty much this motion is going to be just uniform motion. So if you watch what’s happening over here, at every single point in time, if you take a snapshot, what’s going to happen is that these two balls are always going to be at the same height whereas these two balls are always going to be at the same distance, let’s call it, their x-coordinates are going to be the same. So if you notice, the beautiful thing here is, if you are asked the question, I am throwing a ball like this, it’s a horizontal velocity, how long will it take to reach the ground, you don’t have to answer the question. You can find out how long this ball will take to reach the ground and the answer will be the same. So simplify the question. Interestingly if I ask you one more question which is if I do throw the ball like that, how far will it be when it does reach the ground eventually then you know it’s going to be as far as the ball that’s rolling as well. So all you have to find out is – not how far this ball will go, but how far this rolling ball goes and that answer will be same as well. So we have reduced a kind of complex problem into two simpler motions. So even though we knew in our minds theoretically that the horizontal and vertical are independent when we are asked the question about balls falling in the real world our intuition was different, right? So the beauty of this is that it allows us to come up with counter intuitive solutions. For example, if you have a large enough field and you shoot a bullet from a gun and make it such that you somehow release the gun immediately, then from this, what you have learnt here, you can infer that the gun and the bullet will reach the ground at the same time. Now that must be counter intuitive, right? We are of course neglecting air resistance here, but we are doing that all the time. Why is this so? If you fire the bullet exactly horizontally, then both the gun and the bullet have the same initial vertical velocity which is zero. So in case, if you are wondering about if the gun won’t coil back a little bit, even if that happens, if it’s happening along the horizontal, which it will, both of these allow a zero velocity in the vertical direction and both will reach the ground at the same time. Just that, the bullet would have also covered a very, very, very large distance along the horizontal. That’s not going to matter in the question of when it’s going to reach the ground. So the question asked to you was, I shot a bullet horizontally, how long will it take to fall to the ground? You will not ask “What’s the speed of the bullet?”, because it doesn’t matter. All that will matter in this case is the height from which the bullet was shot. Which in this case is 5 meters, let’s say. And the acceleration due to gravity which is 9.8 m/s2. So you will use s equals u plus half at2 where u is zero. You will get an answer, in this case to be 1 second. Of course it’s plus or minus one second but we won’t go there again. Right? Now we saw the impact to the vertical. What about the impact with the horizontal? Before we proceed, there is a little bit of warning, okay? All your intuitions from the real world may not apply really well in the questions that we ask you because in the real world there is some air, some friction, so your intuitions are corrupted by the fact that they exist.
So let me show you why I am saying this. Let’s imagine somehow miraculously on a road there is a lorry going at a constant velocity, right? And why am I saying a lorry? Because it’s usually open. So there is a boy standing on top of it and he has a ball in his hand. And in his point of view he is thinking that he is throwing the ball up, right? So what he does without realizing that the lorry itself is moving, he throws the ball up and with the speed of 20 m/s. The question to you is, how far behind the lorry will the ball land? Now let’s look at this from the ground’s point of view. Now the lorry is going at some velocity and the boy is throwing the ball in a ball that he thinks is up. But he himself is moving along with the lorry in that direction. Which means he has the same velocity as the lorry, which means his hand has the same velocity as the lorry. So just at the moment that he releases the ball, not only is he giving it a vertical velocity as he thinks, but he is also giving it a horizontal velocity. So from anybody who is watching from the ground, the ball is being launched at an angle, with some horizontal and some vertical velocity. But now, do we really care about what that horizontal velocity is? Because the lorry’s velocity horizontally is not really changing. Why? By god’s grace and also because of the facts given in the question. So the lorry’s velocity is not changing. The ball, once it’s launched, its velocity is not going to change because gravity is only acting downwards. So when it finally does come back the lorry and the ball have covered the exact same distance. But not only is that true. Throughout the motion, the lorry and the ball are exactly at the same point horizontally. Which means that the ball is right above the lorry throughout the motion. Now the beauty here is that from the boy’s point of view, he is completely unaware, he is blissfully unaware in fact of all the complexities of what he just did. For him, it just looks like the ball went up and came down to his hand. Now this fact that the boy does not see anything unusual is a lot more important than it might seem at first. In fact it forms the basis in one way for Einstein’s Special Relativity. Now you will get a crystal clear picture of what I am trying to say in our video about frames of reference. We will really play with these ideas, but for now let’s do an interesting application of what we just saw.
Have you ever watched a person skateboarding and there’s a hurdle. They just jump off and beautifully land back on the skateboard right? After they clear the hurdle. Now with this knowledge that you have what can you infer about how much to jump to make sure you land again. Now you know that if the person ends up jumping straight up, so they are neither jumping left or right, then what must happen? They have the same horizontal velocity when they jump as the skateboard’s velocity. Neither the velocity of the skateboard, nor your velocity will change as long as we can neglect some kinds of friction on the board, then what must happen? You will land right back on the skateboard. So for a change this phenomenon that you observe is a lot easier to do than it seems like. Because all you have to do is jump straight up and somehow physics will make sure that you land back again on the skateboard.
Now, let’s take this one step further. Let’s imagine that there is an aero plane trying to bomb a city. If that is too violent for you imagine that they are dropping food packets. Physics of course doesn’t differentiate between these two cases. Then the question is the moment the bomb has been dropped from the aero plane what will the pilot see? The options being these. What is the answer? Option (d) of course right? See I had to stick it in their because of the arbitrary convention that multiple choice questions must have four options always, right? And the only other things I could think of could not be determined and or none of these or some cliché like that. But taking that away, let’s look at the case where the aero plane is flying horizontal. At the moment it drops the ball, or the bomb, or a packet, the meaning of the word drop is that it doesn’t give it any other velocity. So the moment the packet has been dropped, it has the same horizontal velocity as the plane and neither of these change, which means that throughout the journey the packet is going to remain right below the plane which means if the pilot does look down he will see the packet right below him. Now option (a) in that question was there just to see if it still applied intuitions from the real world where if you threw away something from a train, it whizzes back and that’s the kind of intuition we have, where if we throw something from an aero plane, we expect it to go behind. But the fact that that’s corrupted by the fact that there is air is something we try to uncover with that option. So if we take option (a) away as well. Then you have option (b) and (c), which, if you notice, option (b) could have been told to be correct if it did not have the word ‘only’ in it. Which makes option (c) a stronger option. Right? So we are now forced to check whether option (c) is right. Now what will happen if the aero plane is flying at an angle? So, but before we even go there, right? Of the general case with an angle θ, let’s look at a very special edge case. So we can disprove one of the options. What edge case can I think off? So, I want something that is not horizontal. I will pick up the vertical case. Let’s say for some weird reason the aero plane is flying vertically up. If it would have dropped the packet, without a doubt the packet will also remain in the same vertical line. Can’t go anywhere else. So even in that case the packet always remains below the aero plane. Which means option (b) must be wrong, because it says this is true only when the aero plane is flying horizontally. So we have already given a counter example. You know a counter example is enough to disprove something, right? So yes (b) is wrong so (c) must be right, but don’t run away. Let us still learn what we need to learn. So then, let’s take the case where the aero plane is flying at an angle θ. General angle. Then what will be the horizontal velocity of the aero plane? It will be, if the aero plane’s velocity is v, it will be v cos θ. And if the ball has been dropped it will have a velocity the same as the aero plane which is also v, which means the horizontal velocity of the ball will also be v cos θ, so we are beginning to observe the fact that if the aero plane’s angle changes doesn’t really change the fact that both of them have the same horizontal velocities. So no matter what angle the aero plane is flying in, the ball or the packet or the bomb is always going to remain right below the aero plane till it lands. So there is a very interesting implication for all that you have seen here. Have you heard of a game called Angry Birds? Well if you haven’t, go and play that for a while. In fact we are thinking of making an Angry Bird space an assignment before taking gravitation chapter. Now in that, one of those levels there is a white big bird, right? Which drops eggs and kills these pigs. And I was excited. So when the bird was going up you are supposed to click and it will drop the egg. And I thought it’s going to drop the egg. So as the bird was flying somewhere over here, I clicked so that I was hoping the egg would go like that and hit. Which is what you would expect. But I clicked and what Angry Birds did to me was the moment I clicked, the egg fell right down and I felt cheated. I assumed some things and I suddenly felt that the Angry Birds had cheated me by not following the laws of physics very well. And when I was recounting this in class I, one of the new science students told me that “Sir, you made an assumption, do you notice that?” And what was the assumption I had made? I had assumed that the bird was merely dropping the egg which would mean of course what would happen is what I expected – the egg to go a little bit like that. But the bird was not really doing that. And he pointed out a very interesting observation; he said the egg is going down right down which means the egg must have given it some velocity in the opposite direction to cancel out the velocity it would have given it because it itself is moving horizontally, right? So the picture looked like this. Because the bird is moving in some direction like that, if it had just dropped they could have add a horizontal component of the velocity as well. So no the birds has the responsibility to cancel it out. So how should it do that? By pushing the egg in the opposite direction somewhat. Which kind of makes sense because the bird ends up flying that way. Now why is that expected? The bird is applying a force there, it’s getting a force back there. Back in 9th standard, there was taught to us is, Newton’s third law, right? Every action has an equal and opposite reaction. Now the crux here is not to understand that but to understand that Angry Birds does follow some physics. Yes it does. Now you will appreciate this much more when you solve the puzzle in the adaptive flow. When you solve those questions, you reach this question which asks you to find out in which direction the Angry Bird must actually lay the egg, or in other words push the egg so that it ends up going down straight. Happy solving!!! We are playing too much, aren’t we? We are talking about Angry Birds, we are talking about falling balls. We are asking questions about basketballs. But isn’t physics supposed to be serious business? So when exactly do we get to the serious stuff? What kind of serious questions can we ask? If I throw a ball at an angle θ how long would it take to be in the air, how far will it go, how high will it go? Now it’s time to get serious. You know why these serious questions? Because in exams they have asked us the range of projectiles, the time of flight of a projectile and the maximum height of a projectile. And if these questions come in the exam they must be serious right? So in the next view we will see how serious these serious questions really are.