In this video, I show that firing a projectile while aiming at a distance target downrange and dropping it at the same time will always cause the projectile to hit the target, provided that the pro...
Transcript
Let's take a look at Part C. So suppose that the gun is elevated at an angle of inclination alpha in order to aim at a target that is suspended at a height H directly over a point D. So we have a point D here. It's at a height H. So there's the point D. I think this is the point D. So it has a height H here downrange. The target is released at the instant. So it's released at the instant right here. show that the projector always hits the target regardless of the value of v0, provided that projectile does not hit the ground before D. So regardless of the value V0, it's always going to hit exactly here. So we basically find a relation between H. Yeah, before I think I wrote here, but I think it just means H above that point there, yes, it doesn't really have a, unless the point, this is the point D. So anyways, we'll go over the solution. I actually haven't solved it yet, but I'll do it live. Alright, so let's continue further and see what happens in this Problems Plus setup here. Yeah, just because the Solutions Mail doesn't illustrate the answer properly, so it's like I have to literally solve it. real time as I'm filming, and then redoing it and so on, just to wrap my head around it until finally makes sense. So, solution to Part C, if the gun is pointed at target with height H at a distance D downrange, then, yeah, so let's say we are firing, here's our origin there, we're firing here, and then this is going to be where we're firing, that's going to be our height that we're aiming at. That's our height, yeah, so that height is the full age, yeah, so D is a distance there, like this. Oops, here's true that better. So this is the height we're aiming at, and why can't I draw? Okay. Yeah, all right here, I just fix that up, yeah, so we have, this is the right angle. So basically, we're aiming at a target H, this is our angle that we're aiming, firing at alpha, and this is our length, horizontal distance D. So what we have now is then, the tangent, the number is just equals to opposite over adjacent. this equals to just equals to H over D. So that is what the tangent of alpha is. Alright, so going further, so when the projectile reaches a distance D down range, remember, again, first, we are assuming that it doesn't hit the ground first, we have, so it'll lose, I'm going to first go to our trajectory equations. So if I scroll back up, we had it in Part B and elsewhere. Let's just go get at the beginning. So we have this part right here. So there's a trajectory of just a generic projectile being launched. All right. I don't know why it teleported there. But anyways, I'm going to copy this. And, yes, we're going to copy that, and then go right here. Go back to the bottom and then paste it into here. Yeah, so that part B was quite extensive there. and I'm still trying to wrap around my head around Y, so much of that was needed, because it's pretty straightforward. Anyway, so now we have, this is the trajectory right here, so at distance, downrange D, what we get is, I'm gonna just draw this, just so we have a better illustration, so we have our X is our Y, so, and then this is gonna be, we're launching it like at an angle, then it hits this part right here. That's at D. Yeah, so that's our, at D, and this is the initial launch is going to be at alpha, like that. And it's again, we're aiming at somewhere that's H. And I'll just draw that as well. This is drawn fully. Actually, I'll make this a bit shorter. All right here, I just fix it up and made it shorter like that. So that's our launch angle alpha. There's the point that it hits, and then the, I'll just put it, leave it like that. So that's the point it hits in the air. But then this is aiming at somewhere above here. That's where our H is. So that's where it's aiming. It's aiming towards there. So that's our H is the origin like that. So, and then this, we know that this is the X component. So we have this point D. Now I'll put D is equal to, yeah, so we have D is equal to, well, this is our X value, this equals to v0. cosine, and then yeah, this is firing with a V0 velocity V0. So V0 alpha, and then this is going to be, yeah, this is going to be just times it by T, like that. So at the distance, whatever time T it takes to get to there. All right, so what we have is, so I'm going to solve for T. So T is equal to, is move it over to the other side, so we have T equals to D. over v0 cosine alpha. So now that we have that, then we throw this inside there, so in that equation. So we get y is equal to v0, v0 sine alpha, t is going to be, well, d over v0 cosine alpha, then minus g, Yeah, one half, yeah, G, I'll put the one half first. So we have, yeah, so we have one half, G, and then t squared. So we're going to get a D over v0 cosine alpha squared, like that. All right, so then what we end up having is, well, again, as per the earlier part. So we just, the v knots cancel, and then assign alpha over cosine alpha is going to be tangent alpha. So that's tangent alpha, and I'll put the D in front. So we'll put the D in front. Here, just move that 10 in front, so I'm going to put a D in front like that. So, D10, that 10 alpha, minus, and then this is going to be just our, yeah, G, D squared, so I'll put it on top, G, d squared, over, yeah, 2, v0 squared, cosine, cosine, squared alpha. All right, so now meanwhile, the target whose X coordinate is also D has fallen from height H to height, and notice here what we have. When we rearrange this Y equals setup, we have this D10 alpha, which again, D10 alpha is just this right here, so H, so move this D over the other side, we have H is equal to D10 alpha. Yeah, so basically all we have is D10 alpha, and then we have, that's the height here, which equals to D tan alpha, and then this fall, but as we drop it, because number, we're dropping it, and then it's going to hit the launch one, we'll be aiming at it, but then we're going to drop an object, or the target, it's going to go fall down here, but it's going to be falling down due to gravity. Yeah, this is going to be down due to gravity. So it's going to fall down due to gravity at the same rate that this one's falling down to gravity. This object and this one's falling down due to gravity at the same rate. And again, this, and we already have it in this equation, this is just D-10 Theta. So meanwhile, the target, who's this, X-coron, is these also, has also fallen from height, H to height, basically, H-minus, yeah, minus G-T, this one here, G-T-squared. So basically this is the only gravity component. Yeah, so that's just the rate of decrease of, based on gravity. So we have one half G, one half, gt squared, which just equals 2. Again, that's just, again, we already know what H is D10 Theta. D10 Theta. Make this, put a dot there, so it's a bit spaced out. So we have D10 theta, yeah, I mean alpha, not theta, minus, this is just gonna be that, yeah, that's just our GT squared that we already just derived it, so 2v not squared cosine alpha, yeah, so in other words, it's always gonna hit the target, it's always hit it regardless, and yeah, here, just write that out, so thus the projectile hits the target, yeah, so what this basically means is, which is quite astonishing if you think about it, so again, ignoring air resistance, If you have this X, this is the Y. So if you're just aiming at a spot like this, at an angle, so there's your alpha, and you're aiming, so you're aiming at this exact spot, right? But obviously because of gravity, this launch is going to go like this, and it's going to go land down here. So I'll draw this, and here's the origin, so it goes like this, you're launching it, and it goes like that, so then this, and then if you're aiming at this target, let's say there's a ball. and then it drops, or let's do it target, and then it will drop, and then basically this exact thing that your projectile that you're launching will hit the target. So you can literally aim, yeah, so you can literally predict what will happen by just aiming at a spot telling someone to drop it, and then at the exact same time, when you launch, it will strike that over here. So that's actually quite amazing. And I actually, I want to write that out here. I'm going to do hashtag amazing. That's actually pretty amazing when you start to think about that. So you can do a lot of a circus type shows and so on for that.