https://youtu.be/pGaP7K4RzCQ
PHYS 1101: Lecture Five, Part Eight
The last topic for the lecture has to do with free fall, the idea of free fall. By “free fall”, we always mean really only that an object is in the air, nothing else is touching it and usually it’s explicitly stated that air resistance can be ignored. Even if it’s not stated, usually it’s safe to assume that that’s the case. With that assumption, usually you’ll at least get a very good estimate of what the answer to the problem is.
Your day to day experience with free fall would be throwing a ball up into the air, throwing anything up in the air, yourself jumping off the diving board and into the pool. The time in which you’re in the air, you’re not touching the diving board, you’re not yet touching the water, that constitutes free fall. So we’re going to do several problems for which that’s the scope. You want to consider just after the object is let go, it leaves the ground, whatever it may be, to the instant just before it reaches the ground, it’s caught by somebody, whatever, when it’s in its free fall motion, nothing else is touching it.
Okay, here is a fact that you’ve probably heard in a physics class before but not really internalized and realized how counterintuitive it is. Galileo was the first to realize, or propose, that all objects that fall have the same downward acceleration. They all fall at the same rate and, in fact, they immediately start to speed up or accelerate downwards when an object is dropped, it has acceleration. It doesn’t fall to the ground with a constant velocity, it accelerates down. And that acceleration, for all objects, is 9.8 meters per second squared.
I’ve written here a has a value of minus 9.8. That negative is there only because, with the convention of a positive direction being up, the acceleration vector has to point down. The magnitude of the acceleration due to gravity, we call that little g and that’s always just a positive number, 9.8 meters per second squared.
This acceleration has to do with the planet that you’re on. For the surface of the Earth, the acceleration is 9.8 per second squared. It would be quite different on the moon or another planet. We’ll learn more about what sets this later in the class, what determines that number. What does it physically mean, the acceleration, g, is 9.8 meters per second squared, that’s 9.8 meters per second per second.
Said another way, it tells us that every second that an object is falling, it’s velocity vector has to change by the addition of a vector that’s 9.8 meters per second long down. It has to change by a Δv that is downward, that has a length or a magnitude that represents 9.8 meters per second. That’s roughly 10 meters per second, roughly 20 miles per hour, every second the velocity has to change by about 20 miles per an hour. The direction of this Δv is always down.
As examples, this means, if at one instant in the free fall, I have a velocity that’s up and schematically has this type of a length, my acceleration due to gravity says that the next second later this vector added to this is going to represent what my velocity looks like. If you take Δv, so two v0, I’m going to add that Δv. The resultant sum of that v0 plus Δv, tail to tip addition, gets me my next velocity. And that’s what I show here. That’s the tail to tip addition of carrying out what I saw in this equation or what’s shown in that equation.
Okay, or as another example, if an object is already headed downwards and let’s say that this is its velocity vector, after 1 second of falling under the influence of gravity, I’ve got to stretch that velocity vector to something larger. When I add a negative Δv to an already downward velocity, I end up with an even larger downward velocity. So that’s my v0 plus my 9.8 meters per second to give me my next velocity. That’s physically what this means.
What are some of the consequences of that? First important point that is very counterintuitive to people, heavier objects do not fall faster. It’s simply not true. If all objects, if they accelerate downward the same, they’re going to fall at the same rate. It’s going to take the same amount of time to have them fall.
This is counterintuitive because often lighter objects tend to, they’ll have a stronger influence of air resistance than heavier objects, or bigger objects, heavier objects. So our intuition is that lighter objects take longer to fall, but if you take air resistance, the air molecules bombarding these objects as they fall toward the ground, take that out of the picture, all objects fall with the same acceleration. They take the same time to hit the ground.
Here’s a movie I want to show you, a NASA recording that demonstrates that. So, from NASA, it’s a clip here showing an astronaut on the moon and he’s going to drop a hammer and a feather at the same time. On Earth, we all know that if we were to do that experiment, the feather would gently flutter to the ground and take a lot longer to reach the ground than the hammer would. But the argument that Galileo would make is that that’s just because of air resistance, that if you could do the experiment in vacuum where you have no air molecules to interfere, only gravity is the influence that is causing the objects to fall, they would fall at the same rate.
So I’m going to play this a couple times for you, it’s a little hard to see but here’s the hammer, here’s the feather and there you saw the two fall and hit the ground at the same time. Unfortunately, my computer’s not recording the audio that goes with it, but the astronaut is saying, “How about that? Galileo was right in his findings, his predictions, after all.”
Let me play it again for you. Hammer and the feather. They’re on the moon so there’s no air resistance and they do both land at the same time. Okay. So there’s a movie to show you that it’s true.
Here’s a fun little demo that you can do on your own to convince yourself that’s it true. It’s a little bit surprising. If you’ve got a Post-It handy or rip a little square corner off, about a Post-It size off a piece of paper, if you stand up and you drop your pencil and that little square of paper at the same time, that’s the analog of dropping the hammer and the feather on Earth. Air resistance is significant on Earth, thank goodness, that’s why we can breathe and live, and it has a strong influence on how long it takes this little piece of paper to flutter down and land on the ground.
Repeat it again, though, by crinkling up this piece of paper and wadding it up into a tight ball. It’s obviously still light to you. You haven’t changed its mass, but, by wadding it up into a tight ball, you will significantly reduce the air resistance. There just isn’t as much area for the air to bombard and slow it down as it tries to fall. Crinkle up that piece of paper, stand up, and drop it again and you’ll just immediately see that they both land very close to the same instant. You probably won’t be able to tell the difference. So internalize that.
Here’s my little motion diagram to show you. Crumpled Post-It. Pen, something heavy, a book. If you really crumple it so that air resistance is out of the picture . . . let me write this here. So there’s minimal air resistance impact. This means that gravity is really the only thing influencing the motion of this object. If gravity alone is considered, all objects, regardless of their mass, fall with the same acceleration. That means it’s going to take the same amount of time for them to hit the ground.
So this a is 9.8 meters per second, every second, about 20 miles per hour is the velocity change every second, and that’s true for the book as well or would be any object, a heavy object. And of course it’s down. I’m going to write that here. Let me get a thicker marker here to emphasize. Same. Mass not important.
I’ll emphasize why that is in a couple of chapters from now when we get into what causes this downward acceleration, this acceleration due to gravity. Here are a couple bullets just to emphasize what the implications are of this that may not be obvious. I’ve already said this first bullet here: all objects regardless of mass take the same time to fall regardless of mass or size. And, of course, let me emphasize that this is with no air resistance.