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(II) Two Earth satellites, A and B, each of mass $m=950 \mathrm{kg}$ ,are launched into circular orbits around the Earth's center. Satellite A orbits at an altitude of $4200 \mathrm{km},$ and satellite $\mathrm{B}$ orbits at an altitude of $12,600 \mathrm{km}$ . (a) What are the potential energies of the two satellites? (b) What are the kinetic energies of the two satellites? (c) How much work would it require of change the orbit of satellite A to match that of satellite B?

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A) $-3.6 \times 10^{10} J$ $-2.0 \times 10^{10} J$B) $1.8 \times 10^{10} J$ $1.0 \times 10^{10} J$C) $7.9 \times 10^{9} J$

Physics 101 Mechanics

Chapter 8

Conservation of Energy

Work

Kinetic Energy

Potential Energy

Energy Conservation

Moment, Impulse, and Collisions

Cornell University

University of Michigan - Ann Arbor

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Lectures

04:05

In physics, a conservative force is a force that is path-independent, meaning that the total work done along any path in the field is the same. In other words, the work is independent of the path taken. The only force considered in classical physics to be conservative is gravitation.

04:30

In classical mechanics, impulse is the integral of a force, F, over the time interval, t, for which it acts. In the case of a constant force, the resulting change in momentum is equal to the force itself, and the impulse is the change in momentum divided by the time during which the force acts. Impulse applied to an object produces an equivalent force to that of the object's mass multiplied by its velocity. In an inertial reference frame, an object that has no net force on it will continue at a constant velocity forever. In classical mechanics, the change in an object's motion, due to a force applied, is called its acceleration. The SI unit of measure for impulse is the newton second.

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are Two satellites and B are launched into orbit around the earth. On DSO part, I asked for potential energy gravitational potential energy Ah which, as we know, is given by this formula You grab technology is minus G gravitational, constant times, big mass which would be massive, different times small mass which big massive one of the two rockets 9 50 kilograms in both cases over R and R is the total distance from the center of gravity. So so be it. If you're on the surface, this is the radius of the Earth. But these satellites were launched at a certain height, uh, launched into orbit at a certain height above the Earth savoured factory in here too. So I would be our subdue V or radius of the earth, plus the height of whatever satellite. So ah, potential energy off a of satellite is minus G m E times massive a over Artie, plus the height, um, the altitude of it. Ah, and so that's negative. 6.67 times 10 to the 11 10. A negative 11 in s i u nits times mass of the earth, which is 5.98 times 10 to the 24 kilograms times massive the satellite 9 50 kilograms over radius of the earth, which is 6380 kilometers for that 6.38 times 10 to the sixth meters. Uh, plus the height it reaches assaulted tube, which is 4200 kilometers. So that's 4.2 times 10 to the six meters. Uh, therefore, therefore, we have, uh, but the strategy of a of negative 3.6 roughly negative. 2.6 times 10 to the 10 Jules. Okay. And similarly for ah, bee satellite be Ah, it's G tons M e times massive b um, well said G times and me. We know it's 6.67 times a five point 98 types of 24 um, times mass of the massive B M B, which is 9 50 kilograms over a radius of the earth again. 6136138 times 10 to the six meters. Plus this altitude here for B is 12,600 kilometres. So that's 1.26 times 10 to the seven meters. Okay. And so that gives us ah potential energy of negative 2.0 times 10 to the 10. Jules. All right. Um, part B. We want Catic energy, and we don't We don't know the velocity of these rockets that we can just use won't have them be square. Instead, what we do is we equate Ah, centripetal force. I'll call that half subsea to the, uh, to the gravitational force. Ah, which is have some cheesy, um and so centripetal force is just a small mass mess of the rocket. Times v squared over r ah, and gravitational force is negative. Judy times. Ah, small am times, lips, times, big m, Which would be which would be a massive era, um, over r squared Samarkand. So one of the artist canceled, Uh, m small arms cancel. And what you're left with is V squared because she over our on. So that's what we call the escape velocity of the earth. But, um, that note here that if you multiply both sides by 1/2 Ahem, What you get is 1/2 m v squared. Because I hope so. Negative side There is equal 1/2. Uh, negative 1/2 g m the time small hem over our, um and and so this is um, this is actually and so what happened? B squared is kinetic energy and therefore a Connecticut to do kid be is, um this stuff it's negative 1/2 of you because GM you time small. I'm over our is, as you'll recall, that's just your potential energy. So kinetic energy is negative on half potential energy. By the way, this is called the very lithium. Um, but anyway, this is how you can estimate kinetic energy for A and B and self kinetic energy of a is just negative 1/2 times 10 Challenger bay. And that was negative. 3.6 times 10 of the tents. And this is negative 1/2 times they have 2.6 times 10 of the 10 Jules, so can it. Energy of a is just 1.8 tons. 10 to the 10 Jules lips tendrils and similarly kinetic energy be is negative 1/2 potential energy of B. And so that's negative. 1/2 times negative. 2.0, times 10 of the 10 giving us one times 10 10 10 draws. Okay. And finally, in part C, we used, uh, work energy with their, um okay. Writing work, energy And so basically, the idea is that you want work done to change your bed. Let's call a W orbit. Ah, and that will be recalled to the change in kinetic energy, plus the change in potential energy. Right? And so that's Kate Times it's K B. Can you be minus current energy of a plus? You b minus new air, right? And so we can just plug in the values we found in the last two parts. So this is one point of times tend to the 10 minus 1.8 times 10 to the 10 um, and and the other thing. And the other one is minus 2.1 times 10 to the 10 um minus minus. So there's a plus 3.6 time scent of the 10. Everything is and jewels here. Um, and so where you get is negative 0.8 times 10 to the 10 or a talking negative eight times out of the nine plus 1.6 times 10 to the 10 and therefore we get, uh, a 0.0, times 10 to the nine jewels approximately for the work needed to change orbit

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