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Two satellites are in orbit above the surface of the Earth; the first at distance MU kilometers (above the surfuce of the Earth), and the second at distance of 11,2...

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Two satellites are in orbit above the surface of the Earth; the first at distance MU kilometers (above the surfuce of the Earth), and the second at distance of 11,250 kilometers (above the surface ofthe Earth).Assume, for the purposes af the exereises that follow. that each satellite i> in stable (circular) orbit; that the mass the Earth (approximatelv) 10" kilograms. that the average radius of the Earth (approximatelv) 6,38 10* kilometera_Determine the linear (tangential) speed of each

Two satellites are in orbit above the surface of the Earth; the first at distance MU kilometers (above the surfuce of the Earth), and the second at distance of 11,250 kilometers (above the surface ofthe Earth). Assume, for the purposes af the exereises that follow. that each satellite i> in stable (circular) orbit; that the mass the Earth (approximatelv) 10" kilograms. that the average radius of the Earth (approximatelv) 6,38 10* kilometera_ Determine the linear (tangential) speed of each of the two satellites. Present all detail ofthe required calculations. Do NOT round. Determine the period of each of the two satellites. Present all detail of the required calculations; Do NOT round. Determine the e angular speed of each ofthe two satellites. Fresent all detail ofthe required calculations . Do NOT round. ball wilh MaSS 175 kilograms Ad A0 initial cpeed 8.00 mneters per second [20]undergoes hend collision Wilh Gccona ball chat initially at rest Assuming that the collision i8 perfectly elastic and that the nrst ball rebounds with speed equa| OQ meters cccnd determine the mass ofthe second ball. Present all detail othe required calculations Do NOT round bullet with Mus$ of 28.5 Grams, initially traveling at speed o 250 meters per second, [20] becomes cmbedded in a 2.00 kilogram pendulum hanging on light cord measuring 1.00 meter InL lengtn: Upon impact; the pendulum (which now houses the bullet) swings upward Ignoring the mass the cord determine the horizontal component and the verticn] compxnent of the mAximum displacement of che pendulum. Fresent all detail ofthe required calculations Do NOT rund



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A satellite of mass $m$ is moving at a constant speed $v$ around a planet of mass $M$ in a circular orbit of radius $r_{0}$, as measured from the planet's center of mass. Determine the satellite's orbital period $T$ (the time to complete one full orbit), as follows: (a) Coordinatize the orbital plane by placing the origin at the planet's center of mass, with the satellite on the $x$ -axis at $t=0$ and moving counterclockwise, as in the accompanying figure. Let $\mathbf{r}(t)$ be the satellite's position vector at time $t$. Show that $\theta=v t / r_{0}$ and hence that $$ \mathbf{r}(t)=\left\langle r_{0} \cos \frac{v t}{r_{0}}, r_{0} \sin \frac{v t}{r_{0}}\right\rangle $$ (b) Find the acceleration of the satellite. (c) According to Newton's law of gravitation, the gravitational force exerted on the satellite by the planet is directed toward the origin and is given by $$ \mathbf{F}=\left(-\frac{G m M}{r_{0}^{2}}\right) \frac{\mathbf{r}}{r_{0}} $$ where $G$ is the universal constant of gravitation. Using Newton's second law, $\mathbf{F}=m \mathbf{a},$ show that $v^{2}=G M / r_{0}$ (d) Show that the orbital period $T$ satisfies $v T=2 \pi r_{0}$. (e) From parts (c) and (d), deduce that $$ T^{2}=\frac{4 \pi^{2}}{G M} r_{0}^{3} $$ that is, the square of the period of a satellite in circular orbit is proportional to the cube of the radius from the orbital center.

To coordinate eyes. The orbiter we have R. Of T. two vehicle to our notes. Cause T. To I let's I don't know it's a sign. Sita peter G. So you have I hear right out. I bless that's and the distance the just dance shovelled along the circle along the circle in a sign. Seeing she is V. Same C. That is reads times time. This is the reeds times time which records wish equals the circular a claim. Oh no tita Which implies that. See it's uh it's equal to V. T. Divided by our notes. This implies that um So this implies that artsy. It's equal to uh No it's because theater is VT divided by our notes. I legs. Oh no it's sign. We C divided by our notes. G. That is swimming. When we coordinate size their orbits are clean. This is what we have. Then the second part is to be find the acceleration of. There's such lights. We have V. See the vehicle C. D. R. D. C. Which is equal to so have -5 sign. We C divided by our notes. I leads because we C divided by our notes steve. So this implies that e acceleration E is going to be devi over the sea. And this would give me minus V squared divided by Oh no, it's of course we t divided by our no, it's I plus the differential of that is we squared divided by our notes sign. So that is also minus here because if you do friendships close, yeah, getting my next sign. V C divided by our notes jean and this is equal soon minus 50 squared divided by our notes. I have our notes cause we C divided by our nuts. I less. I don't know it's sign. We C divided by our notes. She So this is the acceleration. So this is this will virtually give us mine it's we squared divided by I'm not squared are of C from that because you know what's Rfc is from here, R O C. So hey, this is the celebration of the such lights. Then the nest is to find the force which is equal to M E. And we want to prove that's so see see you have F to be equal to M mm. So this implies that I have flaws to have minus she mm hmm divided by I'm not squared. I have are divided by our notes to be called to em times A we know what A is. It's minus V squared divided by our north squared times. So what do we have here? So I have minus Gm divided by our cube are not cute. Am here councils that and this is going to be equal to My name's MV 2ared divided by our notes squeeze. So this implies that our V squared, it's going to be equal to she am divided by our notes. So what happened? He realized that this and that goods are we then these councils. Does she have one of the guys Sufi, It's going to give us she? M divided by their So then we have been able to show that this is equal to that's then there. Yes, but deal did I say? Let's see is the time is it time for? But there's such lights to complete one full of it's one food updates. So this implies that V. C. It's equal to the circumference of a psycho. So there's a good friends of a cycle is soup. I our notes bangs the last bus. If we substitutes, if we substitute V two vehicle 2 to 5, soup by our notes. Divided by C. See into the squared GM divided by our notes. This implies that we have so This implies we have four pi four pi squared are not squared divided by T squared to be equal to G M, divided by uh nice Sudan howard C. So this implies that r t squared is going to be go to for I four pi squared, I'm not squared are not skewed because this woman supply that so I'm not so cute divided by G I am. So what does this mean? This implies that our C squared is proportional. Yeah, is proportional. Super proportional is directly proportional. Two. Our notes cube says this whole thing is a constance. So since this, since four pi square divided by G M is a constance.

We have angle the moment. Um l equals m r. The sign better now because better equals 90 degree at both the locations we have. L equals m r one V one Prime. It was m are to the two prime. So, um can cancel out. And we want prime. Because we do. Prime times are one are two over our one. So that is one equation. The other equation is the energy conservation, conservation, off energy. We'll give you half am the two prime square minus g m m. Over our two equals half. Mm. The one time square miners G m m over our Juan. Yeah. So there's a two equations and there are two unknown. And solving this you get, we do Klein equals swelled off to G and are one over R two times are ones that are too on. Just look at this and you can solve for V one tribe easily. Which is to G m R. Two over. Our one are ones that are too now putting the numbers in. He will find you on prime. It was 10,000 1 60 meeting for a second or 10.16 kilometers for a second video prime is already found on V two equals quelled off GM over R two on that will come out as 3.7 He let me, dad for a second walk. Done. It was half and we do squared minus f m we One squared equals 3.43 times 10 to the three. 10 to the nine Jules on the total worked on It was 2.513 times 10 to the attend June.

Inbar a off this gushing no more day commission is given as why equal X minus. I mean divided by one night. My lab I X squared. No substituting g equal. They do coin. Oh, be good. Why equal x minus. Did you buy you ride it by one night. X is square on dividing it we get Why is it will do X minus Seattle Point Fetal 168 x is square No again substituting g equal well point seats since for Mars he is equally well, boy. Six we get Why equal x minus. Well, wine six divided by 19 Group wanted lover excess Where? So why is equally true? X minus 0.0 Tzeitel 66 X squared. No, this gets the graph off. Why was X minus 0.168? Excess sweat and why equal ex Lina's? You know wine 0066 X described No the sketch. The drop off y equals X minus zero point c 00168 Excess trail and, like will Exline a 0.66 x discretion in part B off this cushion or no was more than Why is equal X minus tighty through point? You divided by one night X is square No, From this they will get by is equal to minus 32.2 divided by nine in you know, excess grant Minus one night Your I V by so a new point to my deep fly by x no. Here four b is equal doom minus tidy through point. You divided by 19 So no, we get B is equal to minus 4.3 You are guided by one night. Oh, no. Or the second mortal lie is equal to X minus 12.6 divided by one night X square It just it Would it go minus 12.6 divided by one line My deep fly by excess fare minus 19 You are guided by 12.6 more deep level X No here or B is equal. Do minus 12.6. Divided by one line. You from here it will get B is equal. Do minus 3.15 derided by 19 No, no. The difference Origin Total distance traveled by the ruble's is more dealers or capital B minus more dealers off small be it will go 4.3 Do I did by one night, minus jean point 15 Divided. But one night, just a quick ago zeal. Wine 0005 does the difference off origin. Total distance off blue balls is 0.5 mirrors.

For this problem. We're looking at a ball being thrown and moving in a parabolic shaped arch on different planets because the gravity is different on different planets, the distance and the shape of the problem will change from planet to planet in the solar system. And it's the its position of our Balkan be given by this equation where G is the gravity on a particular planet or mood. So we're gonna look at two different planets. First, we're going to look at Earth, and then we're going to look at Mars now for Earth. We're told that G equals 32.2 so I can plug in 32.2 in place of G to get this equation, why equals X minus 32.2, divided by 1922 times X squared. Now, we could simplify this a little bit if we wish we could get rid of the decimal. Um, just factor. It's slightly. I'm gonna leave it like this for the time being. This is how I'm going to graph it. But you're more than welcome to simplify it any way you wish. Now for Mars, we're going to do the same basic thing X minus. But Archie is going to be different from Mars. G is actually going to be 12.6, so that will become our numerator. And again, you can simplify this if you wish. Get rid of the decimal. I'm going to leave it in this format. And let's graph it week, a graph on a graphing calculator or a graphing application on the computer. As you can see, I have graft. Both. The red is the equation, with 32.2 in place of G. So the Red Arc shows what would happen on Earth. While the blue one shows what happens on Mars, where G equals 12.6 light or gravity, you can see from the arc that will travel a lot farther, all else being equal. Now we also want to know what the difference in the horizontal distance is. This traveled by the two balls, So let's look at Earth first. If I look at the point when I am back down on the ground again, where I have no more horrid are vertical distance, I'm back onto that axis on the bottom. You can see that I've traveled for 59.69 ft and I'm gonna come write that down on Earth. I traveled 59 0.69 ft. What about Mars? Will Mars is my blue line here. And if I look at where it comes back down to the X axis, it has traveled 152.54 ft, 152.54 ft. And what's the difference between these? Well 152.54 minus 59.69 gives me a difference of 92.85 feet or almost 93 ft. That is the difference in our horizontal distance that the balls have traveled.


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