7. Friction, Inclines, Systems

Inclined Planes

# Runaway Ramp

Patrick Ford

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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 m 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 got this incline like this. It's kind of like a runaway ramp is, um So I got this truck that's at the bottom here. The initial velocity is 18 m per second and it's moving to the rights. What happens is the truck is going to go up the ramp and it's eventually going to come to a stop. So this V final is going to be zero. 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 V, not we also have a final velocity, so we're having a lot of these motion or charismatics type variables so we can try to solve this using our cinematics equations from trying to find Delta X. I'm going to list out all of my five variables v, not the final acceleration in time. If I can find three out of five that I can find an expression for the Delta X and the reason we can do this is remember that the acceleration on an inclined plane is constant. It's always going to be the same value down the ramp. All right, so we know that this initial velocity is going to be 18 and we know the final velocity is going to be zero. We don't know the acceleration or the time, and that's kind of bad, because that just means that we have only two out of five variables and we need three. So if I want to find Delta X, which is my target variable, then I mean you have I'm gonna 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's no other forces that are acting on our object other than gravity. Remember that the acceleration on an inclined plane is just equal to G times the sine of theta X. So if I can figure out this acceleration using this function or this this equation 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 G right, This is just 9.8. But how do I take the sign of this? 20%. What does that mean? So what happens here is when you are given sometimes these theta angles as percentages, What you're gonna have to do is you're gonna 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 gonna do here is R Theta X is gonna equal the tangent inverse of 20% divided by 100 so really fatal X in terms of degrees, it's just gonna be the inverse tangent of 0.2. And if you work this out, where you're gonna get is 11.3 degrees. So this is the number I'm going to use in this equation here to find the acceleration. Somebody's you 9.8 times the sine of 11.3. Where you're going to get here is you're gonna get 1.92 m per second squared as your acceleration. Now, remember, that acceleration usually gives you the correct sign and the and the direction and indicates the direction here. So the big question is, while this truck is going up, the ramp is the acceleration, positive or negative is up the ramp or down. And if you think about this, if the truck is going 18 and then it goes to zero, that actually means that the acceleration is down the ramp. This is a equals 1.92. So 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 gonna be negative 1.92. So now we have 3 to 5 variables, and we can pick the equation that ignores time. So time is going to be our ignored variable. Here, That's going to be equation number two. So, really, this is the final velocity equals initial velocity plus two, eight times Delta X. There is our target variable. Now we just plug in everything else, right? So this is gonna be zero squared equals the initial velocity 18 plus two 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 gonna get, you're gonna get 3.84 times Delta X is equal to, and this is going to be 324 when you take 18 squared. And so finally, we're going to get is Delta X is equal to 84 groups. This is gonna be 84.3 m. And if you look at your answer choices Wow, this is gonna be 84 m, all right? 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

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