(II) Two Earth satellites, A and B, each of mass m = 950 kg ,&...

Question

(II) Two Earth satellites, A and B, each of mass m = 950 kg , are launched into circular orbits around the Earth’s center. Satellite A orbits at an altitude of 4800 km, and satellite B orbits at an altitude of 12,600 km.
(b) What are the kinetic energies of the two satellites?

(b) What are the kinetic energies of the two satellites?

Accepted Answer

Welcome back. Everyone in this problem, spacecrafts P and Q have been deployed to orbit around the center of the earth. Spacecraft. P is moving at an altitude of 5.5 multiplied by 10 to the third kilometers. While spacecraft Q is at an altitude of 10.8 multiplied by 10 to the third kilometers, determine the kinetic energies of P and Q. If their mass is two multiplied by 10 to the third kilograms. Each assume the orbits are circular. A hint is that the gravitational constant G is 6.67 multiplied by 10 to the negative 11th Newton square meters per square kilogram. The mass of earth is 5.98 multiplied by 10 to the 24th kilograms and the radius of earth is 6.38 multiplied by 10 to the sixth kilograms. Now, what are we trying to figure out? Well, we want to find the kinetic energies of both our spacecrafts P and Q. And in this scenario, what we're saying is that from the earth, OK. We have a spacecraft P that's at an altitude of 5.5 multiplied by 10 to the third kilometers above the surface of the earth. OK. And another spacecraft Q OK. That is 10.8 multiplied by 10 to the third kilometers from the surface of the earth. And we know that the radius OK, of the surface, there are the radius of the earth is 6.38. Well, let me write it in here. The radius of the earth is 6.38 multiplied by 10 to the 6 m. And we have the mass of the earth and the mass for both spacecraft. It says their mass is two multiplied by 10 to the third kilograms each. Now, how can we figure out their kinetic energies? Well, recall that kinetic energy is equal to a half of MV squared. Now, in this problem, we are not told what speed they are moving at. But since the orbit is circular, then we can start to think about some values for the forces that can eventually help us to get to the kinetic energy because we have some equations for our gravitational force and our centripetal force that may include the speed. So let's start to think about think along those lines. No, here, the centripetal force acting on the spacecraft of mass M moving with a velocity FC is equal to MV squared divided by R. OK. Where R is the distance between the earth, the the center of the earth and the spacecraft. So R is going to be the radius of the earth. Plus the altitude of the spacecraft. Likewise the gravitational force acting on the spacecraft. OK, is going to be equal to the gravitational constant G multiplied by the mass of the spacecraft m multiplied by the mass of the earth divided by that same distance R squared. Now, as we said, because this orbit is circular, we can equate the gravitational force to the expression for the centripetal force. In other words, we are seeing that MV squared divided by R is going to be equal to GM divided by R squared. Now, if we think about it, if we divide both sides by two or in other words, multiply both sides by a half. OK. And we multiply both sides by R. OK. Then our equation is now going to become a half of MV squared because R is going to cancel being equal to GM me tmme divided by two R. But no, why did we do that? Remember earlier? We said kinetic energy equals half MV squared. So now we have an expression for our kinetic energy. So our kinetic energy is equal to gmme divided by two R know that we have our kinetic energy, we can use the mass and the distances for both of our satellites to find their kinetic energy. Because no, this means for spacecraft P sorry, not satellites but spacecraft for spacecraft P. OK. Then the kine it's kinetic energy is going to be equal to the gravitational constant 6.67 multiplied by 10 to the negative 11th Newton square meters per square kilogram multiplied by its mass two multiplied by 10 to the third kilograms multiplied by the mass of the earth. 5.98 multiplied by 10 to the 24th kilograms. OK. All divided by two multiplied by R the dis the, the, the distance, sorry R which is the distance between the bodies, the radius of the earth. OK. 6.38 multiplied by 10 to the 6 m plus the altitude of our spacecraft. In this case, for P, it was 5.5 multiplied by 10 to the sixth meters. Now, when we simplify here, we get our kinetic energy to be 33.5746 multiplied by 10 to the ninth joules or approximately 33.6 multiplied by 10 to the ninth joules. Three significant figures. Now let's go ahead and do the same for spacecraft Q. Now for Q that means it's kinetic energy. Then it's gonna be equal to the gravitational constant multiplied by its mass, the same mass multiplied by the mass of the earth, which also remains the same divided by two, multiplied by our distance is the radius of the earth plus our distance for Q which is 10.8 multiplied by 10 to the 6 m. Again, when we evaluate that, then we get it to be equal to 23.2169 multiplied by 10 to the ninth jewels or approximately 23.2 multiplied by 10 to the ninth Js. Therefore, the kinetic energy for P is 33.6 multiplied by 10 to the ninth joules. While for Q it's 23.2 multiplied by 10 to the ninth joules. If we look back at our answer choices, that means A is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Struggling with Physics?

(II) Two Earth satellites, A and B, each of mass m = 950 kg , are launched into circular orbits around the Earth’s center. Satellite A orbits at an altitude of 4800 km, and satellite B orbits at an altitude of 12,600 km.
(b) What are the kinetic energies of the two satellites?

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Intro to Energy & Kinetic Energy practice set

Struggling with Physics?

(II) Two Earth satellites, A and B, each of mass m = 950 kg , are launched into circular orbits around the Earth’s center. Satellite A orbits at an altitude of 4800 km, and satellite B orbits at an altitude of 12,600 km.(b) What are the kinetic energies of the two satellites?

Watch next

Intro to Energy & Kinetic Energy practice set

(II) Two Earth satellites, A and B, each of mass m = 950 kg , are launched into circular orbits around the Earth’s center. Satellite A orbits at an altitude of 4800 km, and satellite B orbits at an altitude of 12,600 km.
(b) What are the kinetic energies of the two satellites?