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Estimate the energy required from fuel to launch a 1465 -kgsatellite into orbit 1375 $\mathrm{km}$ above the Earth's surface.Consider two cases: $(a)$ the satellite is launched into anequatorial orbit from a point on the Earth's equator, and(b) it is launched from the North Pole into a polar orbit.

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A) $5.38 \times 10^{10} J$B) $5.39 \times 10^{10} J$

Physics 101 Mechanics

Chapter 8

Conservation of Energy

Work

Kinetic Energy

Potential Energy

Energy Conservation

Moment, Impulse, and Collisions

Rutgers, The State University of New Jersey

University of Michigan - Ann Arbor

University of Winnipeg

McMaster University

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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All right. So the first thing to note would be to use a fact from problem 52. Ah, namely that the total mechanical energy we call that the mech off a satellite is recalled to negative 1/2 times two times mass of the earth times mass of the satellite over our, um And so so when um so important A Wanna know what, uh, the fuel is or how much energy is. Ah, eyes. His record is required from the fuel to launch the satellite into orbit from the equator. So in this case, you have initially you have kinetic energy. Uh, because you're launching from the equator. Meaning the that meeting, the satellite beating the satellite has on initial velocity because the equator, uh, rotates with respect to deal with the Paul's. Do not certain Part B. We don't have this kind of energy for term eso anywhere here. You have kinetic energy plus energy from the fuel, Uh, which is what we want. Plus, ah, potential energy, gravitational potential energy. A deer, its surface on that is equal to the mechanical energy. All right, so energy from the fuel kinetic energy is 1/2 v not squared will calculate what do not is in a second Ah, energy from fuel as what we need And gravitation Potential injuries Negative G m b time small m over. Are you that being the radius of the earth? And we said that we go to 1/2 g m e have over our, um are is, aren't you plus h rates of us plus height to reach the satellite? Uh, rises. And so the fuel energy from fuel is, um let's take this GMP m term common that times one of our e minus one over two times Ari plus h lips could be an h over there, Uh, minus 1/2 m V. Not square velocity of the earth. Now we're gonna figure fine, v not Savi not will be. Ah, the rotational velocity. So that will be the ah, the circumference of the earth to pi times radius over the period. The time taken for one complete rotation. So this is one day rights. Ah, two pi times radius of the earth which is 6.38 times 10 to the six meters, Uh, over 24 hours times 3600 seconds in each hour What you get is about 464 meters per second. Review. Not awesome. So we have the fuel equals G, which is 6.67 times 10 to the negative 11. Ah, yes. I units has mass of the earth. 5.98 tons. Tend to their 24 kilograms. Uh, times masses of satellite 14 65 kilograms times times one over radius of the earth. 6.338 times 10 to the six meters, minus one over two times radius of the earth. 616 of 113 times 10 of the six plus the height 1.375 times, 10 to the sixth meters. All right, um, And subtract from that negative 1/2 m 14 65 times. Viewed for 64 minutes per second quantities square. Put all of that mess into a calculator or wherever and you'll find Did you from fuel being five point through times? 10 to the 10 jewels are about 54 Giga Jules in part B. The only difference is at the polls. You have no kinetic energy initially? Uh, no. Okay. Initial. Therefore, um, negative G m he, um over R E plus, if you'll is equal to 1/2. Um, negative 1/2 g m e em over. Are you plus h giving us, uh, energy for fuel as, um g m e times m over one over R E minus one over to Ari, uh, plus h. And so, as you can see, the Connecticut do term won't have a view, not square is gone. And what you get is something very close to the last answer. 5.339 times 10 to the 10 Jules, and that's it.

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