Basketballs in Projectile Motion
You know a lot now. In fact you know enough to solve any two dimensional motion problem. But till now we almost always talk about imaginary bodies and some imaginary x, y plane right? Except for a small glimpse about projectile motion. So the question really is, what’s all that we learnt till now got to do with the real world out there? So almost all of us are familiar with basketball. If you’re wondering why I am talking about basketball but I have a ball with FCB written on it with Messi and Neymar on it, because we couldn’t really find a basketball in time. But let’s go with this. So even if you don’t play basketball, you’ve at least watched it being played. Even if you’re the kind of person who believes in conserving energy and not wasting it on outdoor activities like sports. You’ve probably watched this on TV right, while changing channels or something. And if you’ve watched it, you know that most free throws right, almost every single person takes a free throw like this. Right? It’s called the overhand shot. But one of the greatest free throw shooters of all time, the legendary Rick Barry, used a very unusual way of throwing it. He would shoot the ball like this, which is called the underhand, or the granny throw. Now but what particular advantage could throwing it underhand have? Why would he choose that? By the way, the Guinness World record for the most number of consecutive free throws happens to be a staggering 2036, by this guy called Ted Martin. And funny thing is even that wasn’t enough. 1993, Thomas Ambry, when he was 72 years old, broke the record by throwing 2750. And by the way, this particular question, who broke Ted Martin’s record for the most number of consecutive free throws has appeared in competitive exams n number of times. Okay, it’s a really important question, so make a note, the answer is Thomas Ambry, okay. Please make sure you remember that. Now the sad part is he didn’t really miss a shot after that. The gym in which he was doing this had to shut down for the night. And when I read this I was thinking what kind of an idiot was in charge of the gym. It’s almost as if he walks up to this guy and says, “Hey, I know you’ve just broken Guinness World record at an age of 72, but rules are rules. I have people coming here in the morning, they have to do some belly fat and I can’t let a Guinness World record get in the way of that.” Now what happened was, 3 years later in 1996, Ted Martin reclaimed that record with 5221 consecutive throws. Now after that nobody even wants to break the record. It’s almost as if Ted Martin is just waiting there and watching, “Hah, anybody else wants to break it? Dude I want to come and take it again.” It’s pretty much unchallenged, that record. But anyway, I like this. Almost all these people who threw these free throws happened to throw it in a way where when they left, let the ball go, it would go spinning backwards. Have you noticed this? What advantage could that give? Now one more question is, if you watch basketball players, even if there is nobody blocking them, they jump and then take a shot. Why would they waste that much energy? They could just stand and shoot, right? What advantage does jumping give to their chances of scoring a basket? Now these and other questions are what we will be looking at under the name of projectiles, so that at the end of it you’ll be able to answer many more questions.
Now we’ll play with these concepts and get an intuition first. Now whether you’re throwing a cricket ball, an angry bird, a football or even firing a bullet, all these have one thing in common. And that is, the moment they have been released, there is only one thing acting on them, and what is it? Its gravity and it’s pulling them downwards, giving all of them an equal impartial acceleration of 9.8 m/s2. This means that if you do know that something has been thrown, then without even knowing whether it’s a human being, or a ball, or a bullet, you can predict how long it will stay in the air, how far it will go, or, how high it will go. The more interesting fact to notice though is that there is absolutely no acceleration in the horizontal direction, of course if we neglect air resistance, which we usually do. Now, this means that as long as a body is thrown near the surface, we can keep the body’s motion as if it’s a 2-d motion with acceleration only in one direction and no acceleration in the other. But the question then should be how did we even assume that a ball thrown over here is confined to a plane all the time? Or in other words is an example of a 2-d motion? How we sure about that? Now in my class, when I ask this question, right, when I draw 2 axis like this, and I draw a line, a body is like say a ball is rolled along a line like that, and then I ask “Is this motion an example of 1-d motion or 2-d motion?” So one of the common answers I get is that of course it has x-component and a y-component, it’s moving on a plane, so it must be an example of 2-d motion. But the question we really should be asking ourselves is does there exist some line which can completely contain the motion of the ball? And in this case there is, right? That’s the line. Which means that it’s still an example of 1-d motion. The choice of our axis does not alter the nature of the motion itself. Now let’s tweak this example a little bit, and keep our x and y over here and launch a ball at an angle like that. Now the initial velocity of that ball has an x-component, a y and also because it has a height it has a z-component. Then it’s going to go somewhere like that. Now does that mean this is an example of 3 dimensional motion, right? If you think about it, you’ll notice that the question you should be asking is even now does there exist a line that can contain this entire motion? And the answer will be, no there isn’t. So it’s definitely not 1-d motion. But is there a plane? And the claim is, no matter what angle you throw the ball there will always exist one plane that completely contains the motion of that ball, irrespective of how you might choose your x and y axis, and your z. So the final inference is basically this, as long as you’re near the surface of the Earth and as long as you can neglect air resistance, you don’t have to care about at what angle from where the ball was thrown, it’s always going to be an example of motion in 2 dimensions with an acceleration just pointing downwards. But what other assumptions have we made? It’s sometimes difficult to see these. For example we have assumed that the Earth is flat, haven’t we? For most practical throws this is true. Yeah. You can assume the Earth to be flat. So let’s ask this question. Can we use the same mathematics to decide how to launch, say a missile, a friendly missile, from North Korea to the US? Would the mathematics work? So we zoom out and notice, one of the assumptions that we made won’t hold true anymore. We assumed that the gravity will always point downwards but now our definition of down itself will keep changing as we move, because down is really a line that is passing through the center of the Earth. And if the arc we take is long enough, it’s going to keep changing. Now of course we are also neglecting many other things, aren’t we? Yeah. If you’re throwing at a really, really large height, gravity itself is going to be changing. It’s not 9.8 everywhere is it? So, yes, we’ve neglected air resistance, we’ve neglected the fact that gravity changes with height, we’ve neglected the curvature of the Earth. So yes, basically the real world is quite complex as you can see. So we solve our problems in a simpler cartoon version of the world. A cartoon version because some of the rules of the real world don’t really apply there, do they? A cartoon character can just walk off a cliff and not fall down till he notices that there is nothing beneath him. So whenever I see this, almost always imagine gravity just waiting there, waiting for him to see that there is nothing beneath. So perhaps when we are operating in a world that is really complex, a very important art to have is the art of knowing what to leave out in what context. And on that note, let’s see if we’ve left out the right things, because if we have, we can still make reliable predictions about what’ll happen in the real world. So in the next video we’ll look at what skateboarders and bombardiers have to do with what you’ve learnt right now.