 We're now going to work an example problem involving drag on an object. What we're going to consider is a bus that is moving along a surface at a given velocity. So we have a bus that is moving along at a velocity at time and the velocity is going to change. And so what we are given, so we're told that we have a bus traveling at v0. So it's moving at initial velocity v0 and then what we're told is that the engine fails and there is zero rolling resistance between the road and the bus. So all of a sudden the engine fails on this bus and it is going to decelerate and the reason why it's going to decelerate is because there is drag acting on the bus. So the information that we're given about the bus, we're told that the bus has mass m. So we know the mass of the bus is m and it turns out that it has a drag coefficient cd. We know the frontal area of the bus with which the drag coefficient is calculated. So that is given to us. We know the density of air. So it's going through some atmospheric air. I won't put it in infinity, I'll just put rho. So we know the density of the air and I think that is everything that is provided to us in this problem. So what we want to do is we want to be able to find an expression for the velocity. So we're looking for velocity of the bus as a function of time. So the way that we're going to do this is we're going to apply f equals m a to the bus. So let's draw out a free body diagram. Now we can neglect rolling resistance. So the only force acting on the bus is going to be the drag force. And the drag force we know is going to be the drag coefficient times the dynamic pressure times the frontal area. And this is what we're after in the problem, we're after the velocity. So what I'm going to do, I'm going to write out f equals m a. And given the bus is moving in this direction, we're going to take that as being the positive x direction. So the force on the bus is going to be acting in the negative direction. And that would then agree with the bus decelerating through the velocity is going to get lower with time. So what we're going to do, let's plug in our drag force. So we end up with this differential equation here. And remember what we're after. We want to know what is velocity as a function of time. Boundary conditions, velocity at t equals zero, we were told was equal to v naught. So that is what we have to solve this problem. So let's rearrange a differential equation, get dv and v on the same side of the equation. And then let's work through it. So rearranging first of all, before we put dv and v together. Okay, so we've got v squared there. Now we can rearrange. And these are all constants. So I'm going to pull them outside of the integral sign. And on the right hand side, we're integrating. And the limits of integration, time, we're trying to find velocity as a function of time. And we were told the velocity at t equals zero. So we're going from zero to some arbitrary point in time. And the velocity is going from v naught to v, where v is the velocity at some point in time. Which is what we're trying to solve. So what we're going to do, let's integrate this equation. So we get this equation here. Now what we're going to do is just go and do some algebra, rearrange things. So let's walk through that. So we get this expression here. And remember what we're after is velocity as a function of time. So let's solve for that. And so we get this relationship here. So this is the velocity as a function of time of the bus after the engine fails. The only thing slowing it down is going to be the drag force because we're neglecting rolling resistance. And let's plot this just to get an idea as to what it looks like. Look at the extremes that we have for this equation. So if we have t equals zero, we can see that the second term in the denominator goes to zero. And so what we end up with is v of t equals v zero, which is the original velocity. That's what we should have. And another limit that we have, let's say time goes to infinity. As time goes to infinity, the second term in the denominator here is going to get very, very large. And consequently, velocity time equals infinity will equal zero. Asymptotically it's going to approach zero. So what we have are those as being two of the limits in our function. So I'm going to plot velocity and we'll plot time down here. We said this here was at t equals zero. So we begin here v naught and we know way out here we're eventually going to get to zero. And the other thing that we know just looking at this function, velocity of time is proportional to one over time. So it's an inverse relationship with respect to time. So not knowing exactly what it's going to look like, but I'll just sketch it in. Our velocity with time is going to look something like this. So that would be a curve then of our velocity as a function of time of the bus due to drag. Inverse relationship with time. So that gives you an example of a fairly simple problem dealing with drag. But it enables us then to calculate the forces and we can evaluate the velocity of this bus over time.