Hi and welcome to Module 3 of two dimensional dynamics. Here's the learning outcome for today's module. We're gonna go ahead and solve a rectilinear motion problem. So we started to look at rectilinear motion last time. We said it was straight line motion. There was only one component of acceleration and we're gonna call that component in the i direction. So here are the kinematic relationships of position, velocity and acceleration. Since the direction doesn't change, we can work with scalars. And so let's go ahead and solve a problem. So I'd like to get into some real world problems now. And you'll recall back in the introduction video that we saw this picture of this jet airplane landing. Now you won't be able to afford a jet airplane, but you might, at some point, get a small private airplane. And you might want to go ahead and build or design a runway for your small private airplane. And lets say that you're coming into the runway at 60 miles an hour when you touch down. 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're given the initial velocity of sixty miles per hour at touch down. Here is the initial velocity expressed in feet per second since I want to use common units and I have my accelerations expressed in feet and seconds. So now I have the velocity is well expressed in feet and 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 zero, because it's gonna 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 gonna 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 to, since this is an indefinite interval, 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 gonna 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 time 0, which is the same as x dot at time 0 equals 88, which is equal to minus 10 t. Times C1. But, we're saying that t is equal to Zero, so. I'm sorry, this is plus C1. And 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(t) = -10t + 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 then we find out that T is equal to 8.8 seconds to stop. So once I touchdown, it's gonna be 8.8 seconds or once we touchdown in the plane, it's gonna be 8.8 seconds until we stop. Okay? So here we go again. We've got our given information on the right. We now know our velocity is -10t + 88. We know it's gonna 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 traveled. 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 as a function of time, is equal to the integral of velocity as a function of time. Or the integral of, were given now the velocity as -10t + 88dt. Where x(t) equals if I integrate that I get -10t squared over 2 plus 88(t) plus again since this was an indefinite integral, we'll call that constant of integration C2. How do I find that constant of integration? And again you're gonna need to use your initial condition, so we're gonna apply initial condition for the x. For x we have x at time 0. We'll call it our data, that's where the start of our runway is going to be and so it's equal to 0. And so if we 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 now have our total expression for x. So I'm going to write in our mathematical symbol for therefore. x of t equals -10t squared over 2 + 88t, and then I know that C2 is equal to 0. All I have left to do is substitute in the time it takes for the plane to stop. And so I get x(t) = -10(8.8) squared, over 2, plus 88(8.8) seconds. Or the total distance traveled in those 88, or 8.8 seconds is 387.2 feet. And that's going to be the required length of our runway. Obviously we're gonna want to make, for safety factor we're gonna 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 that we saw in our introduction video. Let's say now though 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 traveling at 60 kilometers per hour, and then it's 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 the train will come to a stop at the station, and how long will it take for that train to stop. So that's a worksheet for you to do on your own. I've put the solution in the module handouts, and go ahead and work that. Practice makes you get better and better at engineering problems, and I'll see you next time