[BLANK_AUDIO] Hi and welcome to module three of two dimensional dynamics. Here's the learning outcome for today, today's module. We're going to go ahead and solve a rectilinear motion problem. So we started look at rectilinear motion last time we said it was straight line motion. There was only one component of acceleration and we I'm going to call that component in the I-direction. So here are the kinematic relationships of position velocity acceleration. Since the direction doesn't change, we can work with scalars. And so let's go ahead and solve a problem. So, let's, I'd like to get into some real, real problems now, and you'll recall back in the introduction video that we saw this picture of this jet air plane landing. Now you won't be able to afford a jet air plane, but you might at some point get a small private air plane, and you might want to go ahead and build or design a runway for your small private air plane. And let's say that you're coming in to the runway at 60 miles an hour when you touchdown you're going to decelerate at a constant rate of minus 10 feet per second squared. And you want to find the required length of the runway. So, we are given the initial velocity of 60 miles per hour at touchdown. Here is the initial velocity expressed in feet per second since I want to use common units and I have acceleration expressed in feet in seconds and so now I have the Velocity is well expressed in feet per second. My question to you is, what's the final velocity going to be? And what you should say is, well that's gotta be 0, because it's going to come to rest. And we're given that the acceleration again is minus 10 feet per second squared and it's constant. So how do we find velocity given that constant acceleration? And so what we recall from last module is, we have to integrate the acceleration, so we're going to have the velocity as a function of time is equal to the integral of the acceleration dt, or equal to the Integral of minus 10 dt. And so the velocity with respect to time is equal, since this is an indefinite integral, I get minus 10 t plus some constant of integration, which I'll call C1. So my question to you now is, how do I find that constant of integration C1? And the answer you should come up with is that you're going to have to apply the initial condition. And so, the initial condition is that the initial velocity is 88 feet per second squared. So v at times 0. Which is the same as x dot at time 0, equals 88, which is equal to minus 10 t, times c 1. But we're saying that t is equal to zero, so sorry, this is plus C1. So we get 88 equals C1. So we've solved for the constant of integration, and therefore. We have our velocity which can be expressed now as V of T equals minus ten T plus 88. And so, let's go ahead and look at V final. We said that V final was equal to zero. So V final is 0, when t is t final plus 88, and so we can solve for the total amount of time until we get to a velocity of 0. And we find out that t is equal to 8.8 seconds to stop. So once I touch down, it's going to be 8.8 seconds, or once we touch down in the, in the plane it's going to be 8.8 seconds until we stop. Kay? So here we go again. We've got our given information on the right. We now know our velocity is minus 10t plus 88. We know it's going to take us 8.8 seconds to stop. We still need to find the required length of the runway, so we need to find the distance travelled. And in that case if we have velocity how do I find x. And what you should say is we have to integrate again, and so, X now is a function of time, is equal to the integral of velocity, as a function of time, or the integral of, or given now the velocity is now minus ten t plus 88 dt. Or x of t equals, if I integrate that I get minus 10 t squared over 2 plus 88 t plus, again since this is an indefinite integral, we'll call that constant of integration C2. How do I find that constant of integration? And again, you're going to need to use your initial condition. So we're going to apply, initial condition. For the x, for x we have x at time 0. We'll call it our datum, that's where the start our runway is going to to be, so it's equal to 0. And so, if I put in time is equal to 0 here we find out that 0 equals, this is 0, this is 0, so C2 has to be equal to 0. And so, we know have our total expression for X, write in our mathematical symbol for therefore, x of t equals minus 10 t squared over two. Plus 88 t and I know that C2 is equal to 0. All I have left to do is to substitute in the time it takes for the plane to stop. And so I get x is equal to minus 10 of 8 point 8 squared. Over 2 plus 88 times 8.8 seconds. Or the total distance traveled in those 88 8.8 seconds is 387.2 feet. And that's going to be the required length of our runway. Obviously we're going to want to make for safety factor, we're going to want to make that runway a little bit longer, but it has to be at least 387.2 feet long. Okay? So a simple, particle kinematics problem. Let's look at another real world situation, here's a picture of that train we saw in our introduction video. Let's say now, thought, we have a train that's coming into a station. Here's a picture of a train coming into a station. Let's say it's travelling at 60 kilometers of hour, per hour. And then its brakes are given a constant deceleration of 0.5 meters per second squared. And we want to find the distance from the station where the brakes should be applied so that the train will come to a stop at the station. And then how long will it take for that train to stop? So that's a, a worksheet for you to do on your own. I, I've put a solution in the module handouts. And, go ahead and work that, practice makes makes you get better and better at engineering problems. And, I'll see you next time.