Linear velocity comparison from radius and angular velocity: Worked example | Khan Academy

Let's say that we have two pumpkin catapults. So let me just draw the ground here.

And so the first pumpkin catapult, let me just draw it right over here. That's its base, and then this is the part that actually catapults the pumpkin. So that's what it looks like; it holds the pumpkin right over here. That's the pumpkin that's about to be shot away.

What it does is, once it releases, I guess someone presses a button or pulls a lever, the pumpkin catapult is going to release. This arm is going to turn a certain amount and then just stop immediately right over there. Then that pumpkin, that pumpkin is going to be released with some type of initial velocity, so your pumpkin is just going to go like that.

So this is our small pumpkin catapult. This is our small one. But let's say we also have a large pumpkin catapult. Let me draw that.

So our large pumpkin catapult right over here, similar mechanism, but let's say its arm is four times as long. So that looks about four times as long right over there. So this is the larger one, the large pumpkin catapult. Let me make sure to draw the pumpkin to remember what we are catapulting.

Then it will go through actually the exact same angle; it'll go through the exact same angle and then let go of its pumpkin. This is going to be a very useful video, because you will find yourself making many pumpkin catapults in your life.

So then it will let it go, and you will have some linear velocity. Now we know a few things about these pumpkin catapults. Let's say the small pumpkin catapult; the radius between the center of the pumpkin and the center of rotation right over there, let's say that this is r.

Well, for the large one, this one is this distance right over here is 4r. We also know the angular velocity when this thing is moving. So we know that the angular velocity here, let's say the magnitude of the angular velocity is omega. It would actually be negative if we were to write it as a vector because we're going in the clockwise direction, because that's the convention.

But this right here is the magnitude of the angular velocity. Just to make that tangible for you, we could say let's say that this is, I don't know, 2 pi radians per second. Let's say this thing, while it's in motion, it also has the same magnitude of its angular velocity.

So this thing right over here is also the magnitude of angular velocity, is once again 2 pi radians per second. So my question to you is how would the velocity, the magnitude of the velocity of the pumpkin being released from the small catapult, so v sub small, if I put a vector, if I put an arrow on top of it we'd be talking about velocity; since I didn't put an arrow, we're talking about just the magnitude of velocity. You could think about this as the speed.

How does this compare to v sub large? We have the same angular velocity but we have different radii. Pause the video and see if you can figure that out.

Well, the key thing to realize here, we've seen this in multiple videos, is the relationship between the magnitude of angular velocity and the magnitude of linear velocity. The magnitude of angular velocity times your radius is going to give you the magnitude of your linear velocity.

So for v small right over here, we could write this. We could write this as v small is going to be equal to omega. These are the same omega that omega, and that'll make us the same. In fact, we don't even have to know what this is.

We could say v sub small is equal to omega times our radius, which is r. And what's v sub large going to be? Well, v sub large is going to be equal to that same omega. So I'm talking about this particular omega right over here.

So it's that same omega, but now our radius isn't r; it is 4r. So we're talking about, so times 4r. Or if we were to rewrite this, this would be equal to 4 times omega times r, four times omega times r.

And what is this right over here? Omega times r, that is the magnitude of the velocity of our smaller catapult or the pumpkin being released from the smaller catapult. So just like that, you see by having the same angular velocity, but if you increase your arm length by a factor of four, your velocity is going to increase by a factor of four.

And so you have the magnitude of velocity of the pumpkin being released from the large catapult is going to be equal to four times the magnitude of the velocity of the pumpkin being released from the smaller catapult.