Projectile motion

0:00- [Instructor] Quick question. We have a baseball pitching machine. We use that to shoot a baseball horizontally, and at the same time, we're gonna drop another baseball, okay? Which one hits the ground first? Well, my intuition says the one that is dropped just goes straight down, right?

0:15So it should hit the ground first. But let's see what's gonna happen, ready? Here goes. What do you find? Both of them hit the ground at the same time. Okay, would the result be different if we shot it much faster? Well, let's see. Here goes, boom. And again, the same result, the baseballs hit the ground at the same time.

0:38But why? What does it mean and what can we learn from this? Let's find out. A cool way to analyze things is to first of all look at a slow motion version of it, and then we can take snapshots of the positions of the baseball periodically, say every 10th of a second, and let's see if we can learn something from that.

0:55So let's look at it one more time. We'll take snapshots every 10th of a second, let's say. And let's see what that looks like, ready? Here goes. Boom. Boom. Boom. Alright, so now that we have some snapshots, what can we learn? Well, the first cool thing that we can see is if you look at their vertical positions, they're aligned.

1:20That means when the drop ball comes over here, the shot ball is over here. They're vertically at the same positions. When the ball drop ball is over here, look, the short ball is over here. When the drop ball is over here, the short ball is over here, and the drop ball reaches, you know, the ball hits the ground, the ball that was shot also hits the ground.

1:39This means that vertically, they're both traveling together. In other words, the vertical velocities of both the balls will always be the same at any given moment in time. This is pretty awesome because I already know what the vertical velocity looks like for a ball that is dropped.

1:56We know that the initial velocity is zero, and then as it falls down, its velocity increases, it accelerates, and this acceleration is a constant. Close to earth the magnitude of the acceleration is about 9.8 meters per second square, which we just call as G. Okay, A couple of things over here.

2:14First of all, why is there a Y subscript over here? Well, that's to remind ourselves that these are vertical velocities, and this is the vertical acceleration. We usually choose, well, you know, we usually label the vertical axis as Y. That's why we use the subscript Y over here.

2:30But more importantly, I know the magnitude of the vertical acceleration is 9.8 meters per second square. But what about its sign? I know this acceleration is downwards, but is it positive or negative? Does it depend upon the direction in which the ball goes? That used to be confusing for me.

2:47So how do we decide the sign? Well, the sign purely depends upon your coordinate system. It depends on which direction we choose to be positive. Now, in this example, we have chosen the upwards as positive. Since gravitational acceleration is always downwards, that means in this coordinate system, the sign becomes negative.

3:07So over here, in this particular case, the acceleration would be, in the vertical would be minus 9.8 meters per second square. It has nothing to do with the direction in which the ball is moving. Only depends on which direction we choose positive, okay? Now, we could have chosen downwards to be positive.

3:23That is completely our choice. In that case, the sign of the acceleration, the vertical acceleration would be positive. So it purely depends upon which direction we choose to be positive. And since their vertical motion is exactly the same, I can say the same thing about the shot baseball as well.

3:42Initially its vertical velocity its also zero, and then it increases, it accelerates down at 9.8 meters per second square. But along with the vertical direction, it's also moving forward. So what kind of motion is it doing in the forward direction? Well, let's look at it.

4:01If you look at its spacing, we see the spacing to be pretty much the same, which means the distance its covers in every 10th of a second is exactly the same. In other words, it's traveling with a constant velocity. So the acceleration in the forward direction is zero.

4:20Now, for the forward direction, we can label it as X axis. And so the acceleration in the X direction becomes zero. So whatever velocity with which it was shot over here, that velocity stays the same. Now of course, we're ignoring the effects of error over here.

4:36It does have an effect on both the horizontal and the vertical motion. But if we ignore it, then we can pretty much say that the horizontal velocity stays exactly the same. It's a constant. And now I can combine both these velocities and add them up to get the total velocity of ball at every instant.

4:56So let's do that. And this is what it will look like. So now I can even visualize the path of the ball. This is the path of the shot ball. As it moves forward, its forward velocity is a constant, but its downward velocity increases, accelerates at 9.8 meters per second square.

5:19Okay, now here's a quick question for us. What changes if we shoot the ball slower? How would the path be different? Can you pause the video and think about it now? Alright, let's see. First of all, because it's still shot horizontally, its initial vertical velocity is still zero.

5:38That means vertically, it still has the same motion. Its velocity would still be the same after every 10th of a second, and it would hit the ground in the same time as the drop ball does. But horizontally, since it's shot slower, what happens to its velocity vector?

5:55Horizontal velocity would be smaller, and therefore, in every 10th of a second, it would now cover smaller distance. So if you put that together, this is what the new snapshots should look like, right? And let's confirm that. Let's now look at the animation again, and we can see it's exactly the same as we predicted.

6:19Isn't that incredible? Which means the new path looks like this. So look, the velocity with which we are shooting it horizontally does not affect the vertical motion at all. It's independent of that. If you shoot it slower, well, it'll fall closer. If you shoot it faster, it'll fall farther away.

6:38But again, vertically look, its motion will be exactly similar to the one that is dropped. Okay, so far we analyze what happens to a ball when it's shot horizontally. But what if it's shot at an angle? Like for example, when the batter hits the ball, it's initial velocity is at an angle.

6:57Now, how do we analyze the motion? Well, now what we can do is we can look at this initial velocity vector and resolve it into horizontal and vertical velocities. So we can say it now has this much vertical initial velocity. And this is its horizontal initial velocity.

7:18And again, we can now analyze the motion separately. We can say, hey, vertically, it's kind of like throwing a ball up. Again, we know in the vertical the acceleration is 9.8 meters per second square. And just like before it's sign purely depends upon which direction we choose to be positive.

7:36But when we throw the ball up, what happens to its velocity? Well, as it goes up, because the acceleration is downwards and velocity is upwards, it slows down. So as it goes up, it's slows down, slows down, slows down, it stops, and then it speeds up, speeds up, speeds up, and it falls down.

7:53So let's visualize this. Let's only first consider the vertical motion. Then we'll take snapshots every 10th of a second. You ready? Here it goes. Look, it's slowing down, it's slowing down, slowing down, stops, and then starts speeding up, starts accelerating.

8:08Makes sense, right? During the upward motion the velocity is upwards, but acceleration is downwards that's why it slows down. During the downward motion both the velocity and acceleration are in the same direction that's why it speeds up. But we now know that along with the vertical motion it'll also be traveling forward at a constant velocity.

8:26Which means all these snapshots will be spaced apart horizontally, equally spaced apart. So just like before, if we labeled the horizontal axis as X, the acceleration in the X direction is zero. So let's visualize that now and there we have it. Isn't that amazing?

8:44So we have the vertical velocities. And what about the horizontal velocity? It's gonna be a constant. So whatever was the horizontal velocity over here, the same would be everywhere. And again, we can find the total velocity by combining them. And as a result, we can get the path of the baseball.

9:02So look at what we have done. We have analyzed the motion of an object that's thrown at a random velocity at a random angle. Such objects, which are only under the influence of gravity, we call them projectile. And this motion is called free fall. Even though during the upward motion it's going up, we still call it as a free fall motion.

9:24Now, in contrast, if you consider the motion of a rocket for example, that's not a free fall, it's not a projectile because it's not only under the influence of gravity, there are other forces like thrust, for example, are acting on it. Now, of course, a stone or a baseball thrown in air will also have air resistances.

9:39So they're not truly projectiles. But if we ignore air resistance, then yeah, these are all projectiles and they'll be under free fall. But what's cool over here is if we ignore air resistance and we treat this baseball as a projectile, then it doesn't matter at what angle you shoot it in the three dimension, its motion is always stuck to a plane.

10:00It's moving forward, and at the same time, it's also moving vertically. And therefore, projectile motion is always a two dimensional motion. And both of these dimensions are independent of each other. We can analyze the vertical and the horizontal velocities independently.

10:20And finally, because the accelerations in both the vertical and the horizontal are a constant, they're not changing with time. And we can separately analyze them. This means we can use the equations of kinematics to analyze their motion. In the vertical the acceleration is G.

10:39And as we saw earlier, the sign purely depends upon which direction we choose to be positive. But in the horizontal, the acceleration is zero. So we can just plug that in. And if you do that, notice the first and the third equation becomes trivial. They're basically saying that velocity at any point in time always equals the initial velocity, which makes sense.

10:58If the acceleration is zero, that's what you would expect. But that doesn't help us, so we can get rid of them. And the only equation that survives is this one. So if I ever want to figure out what the velocity or the position of an object is after some time T, I can use this equation to find its horizontal position and its horizontal velocity, which is gonna be the same as the initial velocity.

11:19And I can use these equations to figure out its vertical position and its vertical velocities.