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A cannon fires a projectile at speed V down a hillwith a slope theta.A) If the cannon fires at an angle phi above thehorizon, how far down the hill will it land? 5 ...

Question

A cannon fires a projectile at speed V down a hillwith a slope theta.A) If the cannon fires at an angle phi above thehorizon, how far down the hill will it land? 5 points.B) Show that your answer gives the correct (and famous)result for theta =0. What angle gives the maximum range forthis case? 2 points. C) Would you expect to aim the cannon higheror lower for maximum range if firing downhill?

A cannon fires a projectile at speed V down a hill with a slope theta. A) If the cannon fires at an angle phi above the horizon, how far down the hill will it land? 5 points. B) Show that your answer gives the correct (and famous) result for theta =0. What angle gives the maximum range for this case? 2 points. C) Would you expect to aim the cannon higher or lower for maximum range if firing downhill?



Answers

If the angle of elevation of a cannon is $\theta$ and a projectile is fired with muzzle velocity $v$ ft/sec, then the range of the projectile is given by the formula $$ R=\frac{v^{2} \sin 2 \theta}{32} \text { fect. } $$ What angle of elevation maximizes the range?

In order to solve this question, we will use Thank Person of President Variety Question of prison. The person of Project me, You can write why is equals two extended them minus X. Is going Ladies, man who used them? Prosperity. Who? Here in this question we have Who is the first to be No and access that goes too deep. Let us substitute this in the question of project promotion. We can read why is equals the 10 m minus do the square. Where did we whom we notice? Where possibility let us take for the we see option. Obstinacy is correct.

Projectiles fired up an incline with an initial speed V. At angle theta I. With respect to the horizontal. Where fee is greater than say to I sorry to is greater than feet. As shown in the figure in your textbook show that the projectile travels a distance. D up the incline. Okay so first we have my final is equal to tangent. If data I. And X. Final minus G over to D. I squared. Close and squared of data I. X. Final square. So X. Final is equal to D. No sign of fee. Why final is equal to D. Sign of C. So when we plug all of these in we get D. is equal to two. The initial squared I was gonna say to I fine of data. I percent of the minus sign of fee sign of data. I this is a trig identity over G. Postman squared see so this is trig identity. It is fine of data I minus please. So this is part A. And then part B. Want to find the maximum value. We do that by taking the derivative of the with respect to data I And setting it equal to zero. that gives us an initial angle of 45° of in tangent happens every fee over to it repeats. And then you get C max is V squared one minus brian A. C over G. Co sign square feet.

Okay. So we are given the function in terms of data. So H of data is equal to 12 7 55. Sine squared of pi data over 1 80. So for part a they want us to find each of 20. So this is just plugging in 20° Into your calculator and you have 1492.33. And then from part B we want to find the linear approximation. So I need to find each prime the data. This is 12755. And then the derivative is science too. Sine of pi say to over 180 co sign of pie, say to over 180 Times Pi over 1 80. And so Age prime at 20° Is equal to 1 42.91. So then the linear ization is age at 20 Plus H prime at 20 Data -20 these are degrees. So plugging in and doing a little bit of algebra, You would get 149 2.33 Plus 1, 4 2 9 1 data minus 142 .91 times 20 to a 5 8.2. So this gives us 136 5.87 Plus 1 4 2.91 Data. So if we find inch at 21 degrees, You have 21 pie over 1 80. You have the sign of that number, then squared times 12 755 Make sure you are in degrees.

Former yet a projectile that follows this equation here. See it, players. Because he squared the kinds corruption X over me. Okay, So if we of bringing Thio have around here a hill 200 m high at a distance of 2000 m one of the howitzer. What about you subsea were permitted in the trajectory equation. Well, okay, so when you it's a 100 m height. So that's the value of wine. I don't know what C is. X is going to be 2000 minus e. Think I need to move all this will be a little bit with its whoops. And with that over. All right. Minus one plus c squared Greystone. What's he is? It was 9.8 1/2 times. 2000 over our muscle muzzle velocity, which is 8 97 squared. All right, so we need to find what players of C. I just said Mr. B should be inequality right here. It's to be greater than the head of the bill. Um All right, well, let's get Let's get rolling with this, Then we're gonna simplify all this part right here. Um, what we do, let's see. 2000 over in 97. We square all of that and then remote. Whether by 9.81 divided by two that is equal to 24.38 Um, plus 24 93 8 C squared. This is 2000. See? All right, well, we're almost almost good. Deal right here. Uh, we can subject. Turned it from both sides. Something something a little bit nicer. Hey, c squared. Um, works. They should be the medicines right here. Plus 2000 C minus 224. 24 with 38 Alrighty. All right. And you notice I actually just putting equals and back there, we'll take care of the the minute if I know what the minimum value of C is and what Just, um yeah, we'll go from there. I said from here we can so listen to the quadratic equation that's reminded Square root of 2000 squared minus four times a, um 24.38 conceive. Yeah, already. We're gonna get to five years for see right here. One of them is going to be very correct. One of them is going t o be quite incorrect. Um, let's see What are we gonna hit? Yeah, Swearing minus four times. 38 times. I just really had two bows and see what you want to compete. Native 11 23 One of them is going to get the negative. Anyone? 0.92 Um, I think he did something wrong right there. They're supposed to be positive values. Um, I admit a small calculation here. Here. This is supposed to be negative, right? Our number right here. Supposed to be negative. They are supposed to be positive 0.1123 and 81.92 If we have any bodies that air greater or less than that outside of that range, then basically, what will happen? We'll draw a picture right here. There's a nice little Greenhill. Here's our our Howard Zehr gun right here. Um, if it's too low, it'll hit like that. If it's too high, it'll actually this is kind of go right into everything right here. It's gonna like that, um, and then just won't make it, because any higher than that, So, um, needs to be between this value and this value right here. All right. Syria is going to be um between went 1123 and, um, 81.92 already. So if we want Thio target, that is 75 kilometers a base. That is 75,000 meters. That and, um, what kind of see do we need to have? All right, We already know from this first from party that it seems to be between these two values it doesn't even get inside. There were already We're already toast. Um, everyone. Why to be zero when it hits the target, I believe. And see of C X. Um, thanks being 75,000 minus one plus c squared. Give over to times 35,000 over the muscles. 8 97. Was that what it was? Yes, in 97 and were screened that right there. Okay, looky. Well, let's get so in again. Um, when I first simplify the second half right here quite a bit. Um, all right, this first part of the show, uh, see. All right, uh, 10.8. Over to times 25,000 or anything. You always, uh, trophy the did. Not along with me. Um, it's rather, um, exciting, I guess. Let's see. So we were just with one significant figure there minus, um, that same number times he squared. All right, we gotta throw this into the quadratic equation again. Make sure that our see value is in this range rates here, which we won't know until we actually do it. So let's see. You know, native 75 1000 plus ra minus the square root of 25,000 squared minus four times eight times. See, which is the same things or is just right that Iceland before well, 2 55 squared all over two times you 4255.7 17 Uh, all right, so two values proceed are, can can be between books. I can't take it. Okay, Little rights. And if you have to be a negative number, a cure. All right. So looks like one of our values could be, um, 26. I've and our other value can be one point or for 1.54 pick. Alright. And just to check, these values are between these two values right here. So that means that they both they both work it. Yeah,


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