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Catapult Siege

Patrick Ford
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Hey, guys, let's work out this problem together. So we got this cattle Paul that's launching a stone. It's gonna go up and then it's gonna hit a castle wall. We're trying to get the direction of the stones velocity, so let's just go ahead and stick to the steps. We're gonna draw the paths in the x and Y and then label our points of interest. When you go to do this step, you might actually realize that there's two different ways to draw the path off the stone. One possibility is that if you're gonna hit the castle wall while still on the way upwards, in which the final velocity is gonna be upwards like this or the other situation is that the cattle pump might be launched upwards, reach its maximum height and then actually hit the castle wall on the way back down. And we don't know from the situation or from the information that's given in the problem. Which of these situations is the actual correct one? But it turns out that it doesn't really matter for our purposes, So we're gonna draw the past in the X and y. So the ex path is gonna look the same for both of them. It's just that one of them is gonna be a little bit shorter, but the white pass might look different. The white path of this one is just gonna go straight up and for this one is gonna go up and then come back down again. So when we label our points of interest is actually two possibilities, you could just be going from A to B, in which case your interval is gonna look like this A to B, or you could be going from a to B and then back down to see in which you hit the castle wall. So you're gonna be looking like this A to B to C. So because we actually don't really know, you should always just try to draw the passing the x and y. And if you can't tell, you can't actually figure out which one it is. It doesn't really matter, because we can. Actually, just all we're looking for here is we're just looking for the point which is launched to the point where it hits the wall. So I'm gonna call this interval even though I don't know which one a to see. So we're really looking for Here is we're looking for the angle that this velocity makes with the horizontal. And so remember those two components We've got V, C, X and V C Y. So you've got VC, VC, V, C, X and V C Y. And really, what happens is because we know the X velocity is never going to change. What's really gonna affect this is the why velocity. So let's go ahead and go to the second step here, which is our target variable. What are we looking for? Working for the direction which is data, and we're looking for the direction at point Cease. That's gonna be this variable over here. All right, So because data see is actually just a vector equation, it's just this tan universe over here. I'm gonna start with that. So we have tangent adverse off, and we need V C Y over V c X. So V C X is the easy one because remember that the X component of the velocity never changes. So what I can do here say that my V c X is just gonna be V ex throughout the whole entire motion. And that's actually just the initial velocity times the co sign of data. Right, which I know that's just 50 times the co sign of 56 I get 28. So that's never gonna change. It's always gonna be 28 throughout the entire motion there, V. C. Y is where it gets a little tricky because I don't know what this V c Y is, but I might have to go find it. I'm gonna have to pick an interval and then solve it using some, um, equations. So that's the third step. So we're interval. That we said we're gonna use is just the interval from A to C. So even though we don't know what the actual trajectory is, we don't know if it goes up or comes up and back down again. It doesn't really matter, because all we're really looking for is the point where it's launched and the point where it hits the castle wall. So let's just go ahead and list out all of our variables here in the Y axis. I've got a wife is negative. 9.8. I've got V not why? Which is V A. Y which I can actually figure out here. My v a y is just gonna be, um or rather my v initial. Why is gonna be va y? And that's va V, not times the sine theta. So that's just 50 times the sign of 56. And that is 41 points five man. So 41.5. Okay, so we know this is 41.5, and then RV final Y is gonna be V C y. That's actually what we're trying to look for. And then see, we've got Delta y from A to C and then t from A to C. Now we do know that it takes six seconds for the stone to hit the castle walls. That's t A. C. And so that means that we have 3 to 5 variables and we picked the equation that ignores my delta. Why? So this equation here is actually gonna be equation number one, which says that the final velocity V. C. Y. Is v a y plus a Y times t a c so v c y equals our initial velocity in the Y axis 41. plus negative 9.8 times how long it was in the air. Six seconds. If you go ahead and work this out, you're gonna get negative 17 3 meters per second. So this is our why components of our velocity. Remember, this is not a final answer, because what we need to do is we need to take this number, and we need to plug it back into this equation over here to get our tangent inverse. So what we're gonna do here is Theta C equals the tangent in verse. Off, we've got negative. 17.3 divided by 28. That's the X component. And if you go ahead and work this out, what you're gonna get is you're gonna get negative 31.7 degrees. So which one of our two situations that end up being well, because the y component end up being negative? It's actually going to be this situation over here. This was the correct one. And so this angle over here Thatta see, because it's negative means it's being below the horizontal. So your fantasy is negative 31.7 degrees, and that is your final answer. All right, so that means that brings us to answer choice A. And let me know if you guys have any questions about that one