A comet is in a highly elliptical orbit around the Sun with a period of 2400 years. If this comet has a mass of approximately 5×10145\times10^{14} kg, what is the total energy of its orbit?

−1.24×1021-1.24\$$times10^{21}$$−1.24×1021 J

A comet is in a highly elliptical orbit around the Sun with a period of 2400 years. If this comet has a mass of approximately 5 × 10 14 5\times10^{14} kg, what is the total energy of its orbit?

All right, guys. So for this problem, we have a comet that's in a highly elliptical orbit around the sun. We're told what the period of this comet is, and we're supposed to figure out what is the total energy of the orbit given its mass. So we know we're gonna be working with an energy problem involving orbits. So now the question is which equation are we gonna use? Are we gonna use the energy of a circular orbit or elliptical? We're told that it's highly elliptical orbit around the sun. So that basically just cancels out that first one. We don't have to worry about the energy of a circular orbit. So we're gonna be working with this equation right here. So let's go ahead and write it out. So we've got the energy of an elliptical orbit, actually we don't need part a, is equal to negative g, big M, little m over 2 a. So let's just go ahead and look through our variables to figure out what we know, what we don't know, what we need to find. So we've got g, and we've got the mass of the sun. It's gonna give be given in this table right here. The mass of the comet is given in kilograms right here. But we're not told anything in any information about a, which is the semi major axis. So let's see what's going on here. Remember that when you have a highly elliptical orbit like this, where you have the sun at one of the focus points right here, you have to break the elliptical orbit up into its axis. We've got the major and the minor axis over here, where this variable a represents the semi major axis, and the equation that connects all of these distances together is, let's see. We've got a couple of equations. If you know what the periapsis and the apoapsis distance, so in other words, r a and r p, then we can figure out what the, a is by using the equation r a plus r p divided by 2. The problem is we're not told any of that information. We're not told what r a is or r p. So that means we can't use the equation for this semi semi major axis. We're gonna have to use a different equation. So let's see what else we're told about the problem. We're told the only other information that we have is that the period is 24 100 years. So let's see what else I can use about this problem. I do know Kepler's third law for elliptical orbits right here. And that relates the period of a satellite with the semi major axis, given we know the mass of the thing that it's orbiting. So in other words, we know the mass of the sun, which we do in this problem, and we know what the time it takes, we can figure out what this a is. So that's what we need to use about this problem. So we need to use, the fact we have that we have t squared. So for t sat squared equals 4 pi squared, a cubed divided by g, and then we're gonna write MSun right here. Right? So let's see. If I can figure out let's see. I know what this t squared is. I know what this m sun is, and I'm just gonna move everything else over to the other side. So let's see. I've got this is gonna be t sat squared times g times m sun. Right? Because we're gonna move this stuff over to the other side. And then what happens is the 4 pi squared is gonna go down to the other side of the denominator, and we're gonna have divided by 4 pi squared is equal to a cubed. So if I can figure out what this a variable is, I can plug it back in into the energy equation. So I've got that a is just going be the cube roots of this whole entire expression right here, and all I have to do is just plug in the rest of my variables. So t sats, well, I'm given this is in 24100 years. So the first thing I have to do is I have to convert this. So 24100 years, we're gonna multiply by 365, days per year, and then we have to multiply by how many seconds are in day. I know off the top of my head, it's 86,400. You don't need to know that, and that's per day. So that means that the years will cancel, the days will cancel, and we're just gonna get seconds. So the t in seconds is actually going to be 7.57 times 10 to the 10, and that's seconds. So I just have to plug that in there. So I've got 7.57 times 10 to the 10. Now I'm gonna square that. Now I've got 6.67 times 10 to the minus 11. Now I've got the mass of the sun, which is 2 times 10 to the 30th, and then I've gotta divide this by, and then put this in parentheses, 4 pi squared. You have to make sure that when you put that in your calculators, you're you have that expression, squared because otherwise, the square here might throw things off. Anyway, so you have to do that, and you have cube root the entire thing right here. And then what you should get is the semi major axis of 2.68 times 10 to the 18th, and that's in meters. I'm sorry. That's in that's 13th. So that's not 18. That is 13. So now we have just plugged this all the way back into this equation, and then we are home free. So we've got the energy of this elliptical orbit is negative 6.67 times 10 to the minus 11. Now I've got the mass of the sun, 2 times 10 to the 30th. Now let's see. Let's see if I can make myself some more room over here. So I'm gonna bring this down like there. So then I've got the mass of the comet, which is 5 times 10 to the 14th. Now we've just gotta divide it by 2 times our semi major axis, 2 2.68 times 10 to 13th. And if you go ahead and plug that in, you're gonna get an energy of this total orbit is equal to negative 1.24 times 10 to 21st. That negative sign just means that some of it is potential energy. Right? Energies always have negative signs. Alright. Well, that's basically it for the answer to the problem. Let me know if you guys have any questions.

Physics with Algebra 때문에 힘들어하고 계신가요?

A comet is in a highly elliptical orbit around the Sun with a period of 2400 years. If this comet has a mass of approximately 5×10145\(\times\)10^{14} kg, what is the total energy of its orbit?

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A comet is in a highly elliptical orbit around the Sun with a period of 2400 years. If this comet has a mass of approximately $5 \times 10^{14}$ kg, what is the total energy of its orbit?