Check out this problem here. We've got large trucks on runaway ramps. And what happens is we're told that this truck is moving at 18 meters per second. We're told that this ramp has a 20% grade. We want to figure out how long the ramp should be to bring the truck to a stop. So I'm going to draw this out real quick. Basically, I've got this incline like this. That's kind of like what a runaway ramp is. So I've got this truck that's at the bottom here. The initial velocity is 18 meters per second, and it's moving to the right. So what happens is the truck is going to go up the ramp, and it's eventually going to come to a stop. So this final velocity is going to be 0. What we're trying to do is we're trying to figure out how long, basically a delta x, that this ramp needs to be in order to come to a stop. Right? So if we're looking for delta x, notice the other variables that we have in this problem. We have an initial velocity, final velocity, acceleration, and time. If I can find 3 out of 5, then I can find an expression for delta x. And the reason we can do this is that the acceleration on an inclined plane is constant. It's always going to be the same value down the ramp. So we know that this initial velocity is going to be 18, and we know the final velocity is going to be 0. We don't know the acceleration or the time, and that's kind of bad because that just means that we have only 2 out of 5 variables, and we need 3. So if I want to find delta x, which is my target variable, then I'm going to have to find either acceleration or time. Now remember, on inclined planes, we actually have an expression that we can solve for the acceleration if there are no other forces that are acting on our object other than gravity. Remember that the acceleration on an inclined plane is just equal to

If I can figure out this acceleration using this expression here, I'm just going to plug it back in, and then I can finally find delta x. The problem, though, is that I know

Right? This is just 9.8, but how do I take the sine of this 20%? What does that mean? So what happens here is when you're given sometimes these theta angles as percentages, what you're going to have to do is you're going to first have to convert it to a decimal, and then you can convert it to degrees by using this very simple equation here. If you want this theta angle in terms of degrees, then you just take the inverse tangent of the percentage that was given to you divided by 100. So for example, what we're going to do here is our theta x is going to equal the tangent inverse of 20 percent divided by 100. So really, theta x in terms of degrees is just going to be the inverse tangent of 0.2. And if you work this out, what you're going to get is 11.3 degrees. So this is the number I'm going to use in this equation here to find the acceleration. So I'm going to do 9.8 times the sine of 11.3. What you're going to get here is you're going to get 1.92 meters per second squared as your acceleration. Now remember that acceleration usually gives you the correct sign and the direction. It indicates the direction here. So the big question is, while this truck is going up the ramp, is the acceleration positive or negative? Is it up the ramp or down? And if you think about this, if the truck is going 18 and then it goes to 0, that actually means that the acceleration is down the ramp. This is equal to 1.92. So what we do here is this actually has to pick up a minus sign. So really our acceleration is going to stop our initial velocity, so it has to be negative, right? So this is going to be negative 1.92. So now we have 3 out of 5 variables, and we can pick the equation that ignores time. S_time is going to be our ignored variable here. That's going to be equation number 2. So, really, this is the final velocity equals initial velocity plus 2a times delta x. There is our target variable. Now we just plug in everything else. Right? So this is going to be 0 squared equals the initial velocity 18 plus 2 times a, which is 1.92, and then this is going to be delta x. So, when you move this to the other side, what you're going to get is you're going to get 3 0.84 times delta x is equal to, and this is going to be 324 when you take 18 squared. And so finally, what we're going to get is delta x is equal to 84.3 meters, and if you look at your answer choices, wow, this is going to be 84 meters. Alright. And that's the answer. So that's how long this runaway ramp needs to be in order for the truck to come to a stop.