QuestionTho Initial volocily 0la Car /8 4 Iool per socond ncenlorulta H= volocity ot 1.25 laoVnoa onch aocon Tha Iinal volcly Dia car e 151 i46 por aucond How lorgi...

In Exercises $3-8$ first draw a diagram.
An observer located 3 $\mathrm{km}$ from a rocket launch site sees a rocket at an angle of elevation of $38^{\circ} .$ How high is the rocket at that moment?

In this question, we are given the trajectory of the missile. Okay. Starts from a grows up to be, uh, during that process, the it takes eight seconds. And during the process, uh, the acceleration of the rocket increases linearly with time. And then, um, and he goes to eight seconds. Uh, fuel is cut off and then turns 45 degrees to the right, and then it goes in. Project convention goes, uh, from B to D. Okay, so there are four things we need to find, um, treating. You need to find four things you need to find a speed. True. We need to find the height of the missile HB and the speed of the missile at point B. The maximum higher teen and the range are crashes. That d Okay, so for things to find, So to find that speed and high at point B. Okay, so we look at the graph. Um, the linear line is equal to divide protein divide by eight p, which is I t so d V is equal to a d t. And from zero to be the V c go to 02 t hi T d t and we have three equals to 2.5 T square. Okay, Mhm when TV goes through eight seconds, um is equal to B B is 2.5 times eight square. We get 160 m per second. Okay, so there's this VB 80 fi heights B. Yeah, you know that yes. Is equal to Greedy King. Okay, so from zero to ask. Yes. Secret too. Zero to t 2.5 t square beauty. So, uh, as it goes to 2.53 Yeah, he swear. Thank you. Sorry. Mm. When he is equal to eight seconds, as is equal to 2.5 the right tree. Holmes. It's cute. This is 4 to 7. He does. Okay, so this is HB. Uh huh. Next you can find to the projectile motion. Okay, So for the projectile motion, DVB X. Yes. 160 co sign 45 degrees. So this is actually 82 you know, second, yeah, D B B y. Yes. 169. 45 degrees, which is 82 m per second. Okay, using he's off to find Ah, maximum height. 18. Okay, so this is what How we uh, find it. Okay, so you'll be be wide square goes to B B Y square. Last two, they'll come. Why? So, um Then you take upward to be positive. So zero is equal to e. Uh, two square. My name's too hands. 9.81 Time still talk. Why? A daughter y is equal to 82 square. If I buy two times 9.81 calculate this. You get 652 He does. Okay, So H c is he going to H B s? Delta Y. Okay, so for 27 class 652 Um, yeah. If you use exact numbers, then you should be getting, uh, 1079 km. You run to trees? Significant figures. 51 080 m. Okay, which is 1.8 km. Okay, so this is HC. You can use the 1 m or can Okay, he then to find a range are mhm. Mhm. A range R is able to be the x times t. So this tea, uh, he is the time of flight. So that's the thing we need to find us. It was so to find a time of flight be be using. That's why is he going to Ah, we b y p us have a T square. Okay. And by this time, I'm going to take downward to be positive. Yeah, so I have to. The delta y is h b. It's equal to negative. 82 e bus, huh? 9.81 T square. Okay, so, uh, get 4.905 t square minus 82 e. Right now it's 4 to 6.67 because zero so t using the quadratic formula. Uh huh. K 82 square is 12800 That's four times a. It is minus four a. C. But C is negative. So it becomes a plus. He and the anti by buying 9.81 just to eight and calculate your T to be 26.36 seconds or negative. 3.30 seconds. This is not applicable. So range. Because who were the X? M. T? Which is 80 route two times 26.36 when you get 2980 m, which is 2.98 km. Okay, just so this is the range for the missile. And that's all for this question.

To find how much work is done or required implies that we have F of S Of course to be equal to K divided by X grade. This is guys we have trained divided by K. Uber X. Our exes 4000 squid. This implies that key. It's equal to same times four 1000 scraped. So to find work, W is going to be the NC Gra from 4000 to infinity of saying times 4000 squared divided by eggs square eight. The eggs. And this is equal to their limits. As we approaches infinity we have minus sane 4000 squared divided by eights from uh huh 1000 to be. And this is going to be equal to So if you think the limits us, it's a butch is infinity then I have then 4000. So you have this side minus death side and this will go to zero as it approaches infinity for being it goes to zero, so minus minus I have positive. 4000 squared Divided by eight is 4000. And this is going to give us 400 40,000. Yeah, so this is going to us 40,000 in the unit if Yeah, So we have the units in mouths Hey, milestone for that. Then the second part as well, which is B. It's how far the rockets traveled. We have the total work has has a cake. So how far? So we have the total way When half of that as okay, is divided by two. And that is going to give us 20,000 20,000, which is equally equal to we have my next sane, 4000 squared divided by eggs From was 4000 two. The And this is going to give us -10, both the housing squid divided by B. The last 4000. So if it's so for B so we have to 20,000 to be equal to this this whole thing. So, let me right, it's here. We have this in 3000 To be equal to that. So this implies that you have four 1000 12:10 4000 squared, divided by B To be equal to 20. How was it? So this implies that our B. It's equal to 8000. Yeah. Therefore, the rockets has traveled 40 mouth above itself is so beautiful. Rockets traveled, traveled 4000 mouths above if surface.

In this question, we are given that there's a telescope at this point and a rocket being launched at this point, tita is the anger from the ground to the rocket. Over here. Between the telescope rocket from the ground and let wine be the distance of the rocket from the ground or the high. So let y beer the hype Okay of rocket to ground. Okay. The vertical height. Now I'm trying to find the velocity of the rocket is to find the. Thank you. Changing the Y over change in time. So changing widely. Y over. Change in plan de ti when we are given that teacher at 10 2nd, radiant and Tito at 10.5 It's 0.25 region. So now we actually know what is the real change of the title given these two points here so we can quickly compute that. So um real opportunity to change in data over changing time will be This minus this. So 0.225 -0.205 Over changing the time. This minus this. 10.5 -10. So this gives us to be won over 25 radian per second. Mhm. Okay, lets my equation linking Y and Pita, we know that by tension we can find. So tangent tita Is opposite over Jason. So it's why over four. So let's differentiate both sides. Thank you with respect duty. Now on the left side tangent. Theta Tension. When I differentiate, I'll get 2nd square peter. Now don't forget to differentiate the respect to T. So the teeter over the T. On my right side, one quarter is a constant. I leave one side and I differentiate Y. Respective T. Is just dy over the T. Great. So now I can just stopping all the things that I want to find. So I will have D. Y. Did he is move the fall up over here. I have four now. Second square will be called Science Square deter one overcoat. Science creditor TOm's DTE TO DTE. So I have fall over cool science square. Now that it'll be using is when he goes to 10 will be zero 205 region. Tom's D. T. 30 T. Will be won over 25. So this gives us the velocity is around after I To do the simple places. 0.17 km per second. Yeah.