 We're going to be investigating the motion of this toy fighter jet, which strangely enough is propeller driven. We're going to determine the speed of the jet in two different ways. In one method, we're going to look at the forces acting on the airplane and use Newton's laws and figure out the acceleration and the speed that way. In the second method, we'll measure the velocity directly simply by finding out how far the jet travels in a certain period of time and dividing the distance by the time. And then we'll compare the two methods. So one measurement that we're going to have to have in order to do this is the length of the string that the airplane is hanging from. So let's take that first before we put this airplane back into motion again. I'll just put a meter stick up beside the string. We'll start with it right from the pivot point. And it comes almost down to the airplane but not quite. It takes another, probably a couple of centimeters we'll estimate to get to the middle of the airplane. So that's a total of 102 centimeters or 1.02 meters. Now let's take a look at the theory before we actually take the measurements. Let's look at the forces acting on the airplane from the side. Here we have the weight of the airplane, I'll just call that MG, it's mass times the acceleration due to gravity. And the string pulls on it with the tension force. Well vertically these are the only two forces acting on the airplane. We want to combine these and do a net force analysis in order to determine the velocity of the airplane. And to get the velocity we need to know it's acceleration. So first of all we know that the airplane is moving in a horizontal circle and that when objects move in circles their acceleration is centripetal and is directed to the center of the circle. So the acceleration vector of the airplane is like that. Now it makes sense when doing a force analysis with circular motion to make the axes point, one of the axes point in the direction of the acceleration. So I'll make the x-axis point that way. And the y-axis just has to be perpendicular to that, it can point up or down. So the positive axes are in these directions. Now with those axes let's look at the components of the tension force. I'm going to define the angle that the string makes with the vertical as the angle theta. It also makes this angle equal to theta because these two lines are parallel and those are alternate interior angles. That makes this side equal to T sine theta because that's the side opposite the angle and this side equal to T cosine theta. Now we're ready to write down that force equation. In the x direction we have only one force and that's this one right here, the horizontal component of the tension force. That's T sine theta. And let's look at the vertical forces, the net force in a vertical direction. We have two forces. We have the vertical component of the tension which is T cosine theta and the four weight force which is the other direction so it's negative. Now let's do some physics and algebra to combine these two equations. First we know that the object is not accelerating vertically, it always stays in the same vertical plane. So f net y is zero and we know that f net x is the mass times the acceleration of the object by Newton's second law. So let's write down two equations, ma equal T sine theta from f net x and from the second equation for y I'm going to bring the mg over to the left hand side and have mg equal T cosine theta. Now we want to solve these to get rid of the tension force because we don't have a way of directly measuring the tension force. Well it's easy to solve them that way, we just divide one equation by the other. We'll have ma over mg equal T sine theta over T cosine theta. The m's will cancel and the T's will cancel and what we're left with is a over g equal sine theta over cosine theta. Now there's an identity that says the sine theta over cosine theta is equal to the tangent of theta so the final result taking the g to the right hand side is g tangent theta. So the acceleration of the airplane which is directed toward the center of the circle is g tangent theta. Now that's a centripetal acceleration so that means it's also possible to express the acceleration as the square of the magnitude of its velocity divided by the radius of its circular path and finally to solve for v taking the r to the right hand side and taking the square root the magnitude of the airplane's velocity its speed is the square root of acceleration due to gravity times the radius of the path times the tangent of the angle that the string makes with the vertical. So this is one method that we'll use to find the speed. We'll need to know what the radius of the path is and what the angle is. Now let's think about how we're going to find those things out. What I measured before was the length of the string. Well that's the hypotenuse of this triangle right here. In order to get the angle and the radius then and the radius is right here what we're going to measure is this height. So you see there's a right triangle right here so if we know what this length is and if we know what the height of the triangle is then it's possible to use trigonometry to calculate the value of the angle theta and to use the Pythagorean theorem or trigonometry either one to calculate the value of the radius of the path so we can know all the numbers that we need in order to calculate the speed of the airplane with this method. Now for the other method of determining the speed let's look at this from the top view. This would be the view of someone looking down on the path of the airplane. So here it looks like a circle and the radius of the path is r. The speed of the airplane is v. Now this method is particularly simple it's just a matter of determining how far the airplane goes in a particular amount of time. We're going to use the formula that speed equal distance change in distance over change in time. Since this is a circular path we can use a special formula for the change in distance. We can just use the circumference of the path and the time it takes to go around once c for circumference t for the period of the motion. From geometry we know that the circumference is 2 pi times the radius. So this just results in this formula that the speed is equal to 2 pi times the radius of the path divided by the time it takes to go around once which is the period t. So this will give us a second measurement for speed and we'll be able to compare this then to the first measurement which used Newton's laws and see how well they compare. They should come out to be about the same. Here we are again with the airplane going around in a circle. We need to measure two things the vertical height and the time it takes the airplane to go around. We already have one other measurement that we need and that is the length of the string. So to get the vertical height I'm going to take my meter stick and just put it up to the side here and bring it in as close to the airplane as I can and you can see on the meter stick about where the airplane passes it. Zero is at the ceiling and so that tells you the vertical distance that the airplane is below this point of support. In order to get the time measurement you can simply do that with your real player window because you have a time display in the real player. So what you'll do is watch the airplane and count the number of rotations. I'd say you should count a total of ten rotations and when you begin counting at zero note the time in the real player window and then note the time again after you count to your tenth rotation and so you can take the total time and divide it by ten and that will give you the period. So this gives you all the measurements that you need in order to calculate the speed of the airplane using the two methods that we described.