Example 6: Impact Speed of a Projectile Hitting the Ground by @mes on 3Speak
▶️ Watch on 3Speak - YouTube - Telegram - Summary - Notes - Sections playlist - Vector Functions playlist - MES Links --- In this video, I determine the time a projectile is in the air, the distanc...
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Example 6, impact speed of projectile, and this is also going to be hitting the ground. So your projectile was fired with muzzle speed 150 meters per second, and angle of elevation 45. 45 degrees, you forget the degree symbol there. So 45 degrees from a position 10 meters above ground level, and now we're asked, where does the projectile hit the ground, and with what speed? All right, so it's basically the same thing as the previous example, but the only difference is now that it's being launched from a height above the ground, so 10 meters above, but you're just being launched by it from an angle and so on. So, so solution, if we place the origin at ground level, and then the initial position of the projectile has coordinates 010, so it's 10 meters above the origin. And so we need to adjust the earlier parametric equations of trajectory by adding 10 to the expression for y. All right, and I'll go over that as we solve this. So with, let's write down what we were given. So with, we have V-0. is this initial speed, so it's the muzzle speed, so v0 is equal to 150 meters per second. All right, and now we have alpha, so alpha is equal to 45 degrees, like that, and this just equals 2, well, we know this is just going to be pi, that's just, yeah, pi over, that's 90 degrees divided by 2, so this is just pi over 4, or in other words, pi over 2 divided by 2. Uh, yes, pi over four radiance. I'll put this better, radiance, all right? And we also know that, well, we're going to assume G is equal to 9.8 meters per second, or approximately equal to, yes, fix that approximately sign, or just equal to, doesn't matter, uh, it's all close enough, so 9.8 meters per second squared, so 9 .8 meters per second per second. Alright, and now let's draw, and actually, I'm just going to grow this width part, we don't need to write that, because Kagos Book says with this, then we have, I'm going to move this over here, and I'll fix everything up. All right here, I just move everything over there. So now let's just draw this out, how it looks like the trajectory. So let's say we have the y-axis, and now we're going to have the x -axis. There, and there's the origin, and then there's the height 10 meters. So there's the 10 meters. I'll put the 10 better. All right, 10 meters, all right, so that's 10 meters, and it's going to be launching from like this, and that is our initial launch there, and I'll call this, and it has a V-not velocity, like that, and then there's going to be an angle alpha. Yeah, an angle alpha, like that, and that equals to 45 degrees. or piver or four. So that's just, I'll just put alpha like that. Alright, so that's the launch, and now it's going to basically go in a parabola. It's going to go like this, and then launch all the way down here. So I'm going to move this, actually. It's going to be like that, and I'll put it like that. Yeah, let's make it like this. Yeah, I'll put it above here. Let's see initially it's like straight line, and then it's going to go land right over here, and that is the distance we want. So it's going to land there, and now we want a distance D, and that's what we're asked. Where does it projectile hit the ground, hit the ground, and with what speed? So what's the speed that it hits the ground with? And then we can go V final there. Alright, so let's take a look now at our equations of the parametric equations of trajectory, as I covered earlier. So we have these. So we have, this is the equations. So this is X, what I'll do, this, I'll copy. Yeah, let's copy this whole thing here. So this is what we derived before. And then for the Y, because initially we started at the origin, we just need to add that extra height. So there is our setup there. Yeah, so that is X equals to v .0 cosine alpha at that times T, then this Y is equal to v.0 sine alpha times T minus GT squared over 2. But now we need to add a plus 10, because initially we're starting at the origin. Now we just add an extra 10 meter, so that the initial height's there. So when you plug in zero for time T, the height goes to here. All right, so it goes like that. And now for the X one, well, we know X is going to equal to V -0 is equal to 150 meters per second, 150 meters per second, that's like that, and then cosine alpha, cosine piper 4, and times T. All right, so this is what we have for the X, and we need to solve for X, but there's a T there. So we need to know what T is. and we can solve for t by while setting y equals 0 here so at here we have y equals 0 is with an impact so then we just set this equal to 0 and solve for t and then throw this in there and then we sell for x all right so what we'll do again similar to the other setup so we're going to write now write a y is equal to v0 actually let's uh Yeah, yeah, I'll just put it all here. We're going to have to do some quadratic formula setup, but still, we can just put it on here. So y' equals to v.0 is 150 meters per second, and then times by sine alpha, which is, again, pi of four, because typically dealing with radiance is easier, because all the calculators default is radiance, all right, so that's what we have, times by t. and then minus 9.8 divided by 2, I'll just put that in for completeness, 9 .8 divided by 2, t squared, then plus t, and plus 10. All right, now what we could do is, what I'll do is, well, if you notice here, this sign pi 4, we could get the exact trig ratio of this, so what I'll do is recall, all right, recall, So if we just have a triangle setup, I'll also put it out here. Actually, what I'll do is also put it here as well. So this as well, recall that what we're going to get is, if we were to draw a square, I did a video on this, but I like doing this, like always, instead of remembering, just do it. Okay, so we have this, this is just a square. and its length are 1, 1, 1, 1, 1. And so then the angle in between a cross like this, this is going to be, well, 45 degrees. So that's 45 degrees, in other words, yeah, this is 45. Or in radiance, it's going to be pi over 4. Yeah, so then this one here, using Pythagoras, this is 1 squared plus 1 squared. This equals to, well, just 2 here, this is square of 2. That's a squared 2, so, yeah, just one squared plus one squared and not all square root. That's with the agris theorem, so then this means now, well, sine, sine of pi over 4 is equal to opposite of our adjacent, I mean, opposite of her hypothesis, 1 over square 2. So that's what it is, but also calculus book always puts the squared on the top, so I'm going to multiply it squared on top, and bottom, so aren't changing anything. So what we're going to get is, yeah, so what we're going to get is, well, Yeah, this square 2 times square 2, that's just equals to 2 there. So we're going to get equals to square 2 over 2. All right, so that's what it equals to. And likewise, cosine 5 for 4, this equals 2, well, it's going to equal to, again, adjacent of our hypothesis of 1 over square 2, and that also equals to square 2 over 2. So they're both exactly the same thing. All right, so thus, what we have is, well, the X one is going to be thus, x is equal to, well, 150 times by square 2 over 2, that's just going to be 150, yeah, divided by 2, 75, so this is just going to be 75 square root 2 times t. All right, and then the other one's going to be, Y is going to equal to, I'll box both of them. separately because we're going to have to do is put the quadratic formula on the other one. You know, all right here, I just fix that box up, so this is going to be Y, yeah, this one fixed that up, so Y is going to be, well, 150 divided by 2, is this going to be, well, yeah, 75, yeah, 75 root 2, T minus 9.8 over 2, yeah, minus 9 .8 over 2, that's just 4.9.9. So again, 9.8 divided by 2, yeah, this 2 goes into 9. Well, it's like, think of it, imagine it was 100. 100 divided by 2 is 50, but there's a 98. So 98 divided by 2 is going to be, yeah, 49. Yeah, and then put the dot there, so 4.9. All right, so 4.9T squared, and then plus 10. All right, so going further, so now impact of the ground occurs when y equals zero, as I drew over there, so which we can solve via the quadratic formula from my earlier video and using only the positive value of T we get. So here's a recall from earlier video, there's a link to it, and there it is, the proof I basically use to complete the square, so a x square plus bx plus c equals zero. And then when you set that equals 0, set this one equal 0, and then, yeah, basically, if you complete the square by noting that x plus d, and then all squared, if you foil this out, x squared plus 2dxx, yeah, plus d squared. And then basically you do the same thing with this one here. So you divide A out. So you're going to have a C divided by A, B divided by A, and so on. And then we're going to have to a plus. and minus B, something like that. Yeah, we're going to have to actually, yeah, actually, if I recall, it's going to be B over A, so we're going to have to do something, plus and minus something to get this set up, so that we can just put it in here, so X plus B over 2a, and so anyways, yeah, you can go and figure that out and watch a video for that, but anyways, here is a quadratic form of x equals to negative B plus or minus squared, b squared minus 4, A, C divided by. 2A. So in our case, I'm going to rearrange this so that it lines up with this x squared 2, yeah, x squared x and then the constant on the other side, the 10 on the end. So we're going to have, so I'll write thus, we have y is equal to, and I'll put this negative 4.9, negative 4 .9, t squared, right here. And then this is going to be plus 75 times square of 2. Whoops, square root 2, and then plus 10, and then set this equal to 0. So this equals to 0. All right. Let's fix this up. I keep writing it softly for somebody. All right. Yes, equals 0. So now this one's going to be. our A, this is our B, and this our C. C. Yeah, so we're going to have, instead of X here, our variables T, T, is equal to, well, negative B, which is going to be 75 times it by square root 2, and then plus or minus, and now we have a plus or minus, and then this is going to be, well, square root, B, yeah, this is going to be B squared, yeah, B squared is going to be, well, 75 squared, and then square 2 squared squared is going to be 2, all right, and now, make this thicker, I don't know why it's always so thin, all right, and now this is going to be minus 4a. So A is going to be negative 4 .9, so if it's negative minus negative there, it's going to be well positive. So it's going to be 4 times it by 4.9, and then times it by C, where C is going to be 10. Times by 10, like that. All right, extend it over there, extend it even further like this. All right, all right, this is what we have, then we have to divide by, well, 2a, so 2 times it by A, which is negative 4.9. So I'll put a negative out there, 4 .9, like that. So that's what we have, and now what we have is, well, we have a negative on top, this is negative here, but this is also plus minus. So just divide out the negatives, just be plus, just be plus. But then this one, if it was plus minus, now it becomes minus plus. So if it was minus, it becomes plus, if it was plus becomes minus. But in fact, this is actually bigger than this number, so we're going to get a negative number. So we're only going to look at the positive values. So we only want positive values of t. So what we end up having is, well, this equals 2, just simplify it even further, 75 times square or 2 plus... And then, calculus book just puts the, it calculates this out, 75 square times 2 is actually going to equal to 11,250. 11,250. Yeah, so 11,250 plus 4 times 4 .9, that's going to be 4 times 49. It's actually going to be 196. And I'll do a calculation check soon with my built-in one-note calculator. 196, like that, and I'm going to extend this, all right, extend that, and now divided by 2 times 4.9, that's 9.8. All right, and now this, plug in this a calculator, we're going to get approximately equals to 21.74, and it's actually going to be seconds, just because we were dealing with G, gravity is 9.8 meters per second squared, so we had a seconds, and then we had the 100. the speed was 150 meters per second initially. So yeah, it's going to get canceled, some I'll get S there. All right, so thus, or then, so we have that, then our X value, X, or is it going to be now D, actually. So D is equal to x, x equals to, basically, basically, this is going to be, D equals X, and that is going to be, well, we have this. X is 75 times square 2 times by T now. So this is going to be 75 times square 2 times by T, and now we just throw this inside. So this is going to equal to 75, or approximately now, because that's an approximation. Approximately 75, yes, erase this, 75 times square root 2 times it by 21.74. All right, now if you plug this in the calculator, what we're going to get is approximately equals to 2,306 meters. So at 150 meters per second, yeah, meters per second velocity, you're launching something at 45 degrees, you're going to get basically, it's going to go, it's going to be in the air for 21 seconds and land, and it's going to go 2,300 meters, holy, so 2,306 meters approximately. All right, yeah, so basically, yeah, the projectile hits the ground about 2,306 meters away, and just a quick calculation check. So we have 75 squared, yeah, 75 square times 2, that's this part right here. Yeah, we just put in the one note calculator, just erase this, put equal space, and it goes there, 11-2-50, so that's this one, here, and now this one, four times 4.9 times 10. So 4 times 10, so 4 times 10, that's just, yeah, 196, again, you could press space. and then 11250 plus 196 equals 11446. Yeah, I just put that there, and then the reason is just, if you squared 11446, that's this part right here, we're going to get a 106.98, but then if we have this 75 times square root 2, we get this 106.06.06. It's interesting, so it's a bit off, I'm not sure. Why? I think it's because, yeah, actually, I think it's because it's 4.9, that's gravity, that's more of an approximation, because this one is just 75 times square root 2. So that's an interesting setup there. Yeah, I'm not sure exactly why that's different, but then, but anyways, so if you made the negative there, these would pretty much cancel it. But yeah, I'm not sure what the rounding errors, or it might be just the calculator away with that square root, it might be different. then square root 114, 4, 6, etc. So whatever reason is, but in theory it should be cancel out. So the exact one, these should cancel. So if you have a minus, you get t equals 0. So in other words, we just ignore the negative sign, which do the positive. Yeah, I just wanted to double check that, so to see what happens. And now if you go 75 times square root 2, right here, times it by, this square root 11-446, that's add these up. and then divided by 9.8, so divided by 9.8, this is going to be 21.74, and this is going to be 21, yeah, that's it. Space, so that's it. Yeah, so that's it, and then finally, we have this one here, the distance, D equals X, basically 75 times square 2 times 21 .74. So 75 times square 2 times 21.74, that equals to this whole giant thing, space. So 2305.875, like that. And then just round this up 2, 306 meters. So, yes, fascinating stuff there. All right, and now the next part of the question, that's actually on the velocity of the projectile. Because remember, we're asked to find the distance it goes. So basically, we're asked, where does a projectile hit the ground, and end with what speed? So we need to find the actual speed hitting the ground. And again, so we have the distance right here. That's about, what is it, 2, 336 meters. Now we need to know the velocity. Well, we know the velocity, the projectile is, well, is just a derivative of the position vector. But first, so note, or I'll just say we have, yes, so the position vector is going to be R of T, and this equals to, well, we have the parametric equations of it. That's this part right here. These two are the parametric equation, so let's just put those in their I and J format, so 75 square two. is going to be like this, so it's going to be 75, square root 2 times t, and it's going to be I, like that, and then the next one's going to be plus, and then plus, this one is 75 square of 2, t minus 4.9, t squared, plus 10. So we're going to get a plus 75, square 2 t squared, and then this is going to be, yeah, now we need to put a bracket, actually, and then the next one's going to be minus 4.9, no, this is not t squared, this is just t, that's just t, that's just t, let's go back up, that's just t, and then 4.9 t squared plus 10. All right, so this is going to be 4.9, t squared, plus 10. and this is going to be our J component. And this is going to be bracket J, like that. All right, so we have that. Yeah, so, yeah, I moved it over here. So thus, the velocity vector, v of T, is equal to, well, the derivative of the position vector, R prime of T. So now this equals to, let's just take the derivative, this T, this just becomes just a constant. So we're going to have a 75 square 2, that's it, I. And now this one here, well, the derivative of 10 is going to be 0. This one here is going to be, well, the T just vanishes, so we're going to have plus 75 square root 2, and then, yeah, the T's gone. Next one's going to be 4.9 times 2, that's going to be 9.8 times T. and then J. All right, so that's what we have, and yeah, that's the velocity vector. So its speed at impact is, well, at T is equal to, well, 21.74, so at that, so, and then we need to get the magnitude of that, so the speed is basically the length of it, or the magnitude of it. So, thus its speed at impact is going to be the, this absolute value sign, so of T at, well, 21.74. That's what T equals 21.74. seconds, so it's going to be like that, so then the length of it. And then this one, again, it's just the 3D distance formula, or 2D, doesn't matter. And now this is going to be square root, and then we have to square root, square the components. So 75 squared, it's going to be, well, 75, I'll just put it like this. Yeah, the calculus book just puts it all like this now, 75, square root two, squared, and then plus, now this whole other setup here. There, so we're going to have a 75 square root 2 minus 9.8 times, and then 21.74, times 21.74, and then all squared, like that. Yeah, so that's the dot, make it higher, so it doesn't look like the decimal place, so that's it. Now right here, just moved it over, and I would just extend this over like that. Okay, so that's what we have. And now we plug this into the calculator, this is going to be roughly equal to, well, 151 meters per second. Yeah, this is going to be meters per second, like that. So that's the velocity upon impact. Alright, yeah, that's what we have, 151 meters per second, and lastly, let's do a calculation check, so here's again with my built-in one-note calculator, yeah, I can just write square root, and then have seven, and then put a bracket, 75 times square two, and then square this side, and then 75 times square root two, minus 9.8 times 2 .8 times 21.74, and then square root that, and then put the bracket at the end. Again, I get you press equals space, and I get that whole way. So yeah, 150.65, rounded up to 151. So, yeah, good enough. All right, so that's an absolutely fascinating.