A space station orbits the sun at the same distance as the ear...

Question

A space station orbits the sun at the same distance as the earth but on the opposite side of the sun. A small probe is fired away from the station. What minimum speed does the probe need to escape the solar system?

Accepted Answer

Hey, everyone in this problem, a spacecraft is located at a distance from the sun equal to that of Jupiter's orbit and it needs to escape the solar system. So we're asked to determine the minimum speed for the spacecraft to escape the gravitational pull of the sun. We're told to consider the distance between the center of the sun and the spacecraft rs to be 69.9 times 10 to the exponent 6 m. We have four answer choices all in meters per second. Option, a 3.79 times 10 to the exponent six. Option B 1.95 times 10 to the exponent six. Option C 1.95 times 10 to the exponent 12. And option D 3.79 times 10 to the exponent 12. All right. So when we think about this problem, let's consider the sun as a spherical mass and the spacecraft as a particle. When we do that, those two things form an isolated system, which means we have conservation of mechanical energy. And so now we call, if we have the conservation of mechanical energy, we're just gonna short form a little bit here. Well, that tells us that the initial mechanical energy emac one is going to be equal to the final mechanical energy or the mechanical energy in the second state. So we have the initial is equal to the final mechanical energy. And we're called the mechanical energy is the sum of the kinetic energy and the potential energy in the kinetic energy we can write as K, so we have K one plus the potential energy, which we can write is U. So K one plus U one is equal to K two plus U two. All right, let's expand a little bit here. The kinetic energy recall is given by one half MV squared. OK. We're talking about the kinetic energy. We have the spacecraft, OK. This is the spacecraft that is moving. So when we have the mass there, that's gonna be the mass of the spacecraft. So we have one half, we're gonna say little M subscript S is the mass of the spacecraft multiplied by V one squared. We're gonna add our potential energy and in this case, the potential energy is gonna be gravitational potential energy and we're not dealing with springs or anything here, but we do have that gravitational pull from the sun. And so our gravitational potential energy here is going to be negative, we're going to subtract, going to be capital G multiplied by the mass of the sun, which we're gonna say is big M sun multiplied by the mass of the spacecraft MS divided by R one Newton's law of gravitation. All right. So we've done the left hand side of our equation. The right hand side is gonna be the exact same. We just have different subscripts. OK. So we get, we have one half the mass of the spacecraft MS the two squared. OK. That second speed minus G, the mass of the sun, the mass of the spacecraft all divided by R two. OK. So that distance or the radius in that second scenario. All right. So this looks messy. We have a lot of terms here, but we're going to s now recall that as the spacecraft escapes the gravitational pull of the sun, the speed is going to be reduced to zero. OK. So as the spacecraft escapes, what's going to happen is the distance that radius R two is going to be going towards infinity. OK. We're getting further and further and further away from the sun and the speed V two is going to be going off to zero. And this is the condition that we always use when we're talking about escape speeds. All right. So if we have V two going off to zero, then the right hand side of our equation, that first term is gonna be going to zero. And if we have R two going off to infinity, we're dividing by R two. And so the term with R two and the denominator is also going off to zero. So what we're left with, once we apply that condition is that one half multiplied by the mass of the spacecraft multiplied by V one squared minus capital G multiplied by the mass of the sun multiplied by the mass of the spacecraft divided by R one is equal to zero. All right, in this speed, the one that is our escape speed and that is the speed we are looking for. So let's rewrite this. We're gonna move the second term to the right hand side by adding it. And we're just gonna rewrite this to be our escape speed. So that it's clear we have one half multiplied by the mass of the spacecraft multiplied by this escape speed squared that we are looking for, that's gonna equal G, the mass of the sun ma of the spacecraft divided by R one again, that's capital G there. Now, we might be looking at this thinking, well, we know big G and we have the mass of the sun. That's something we can look up in a table in our textbook. We were told this radius and we were told this our value because we were told the distance from the center of the sun and the spacecraft. But what about the mass of the spacecraft? And we weren't given that however, we have MS on both sides, we can divide both sides and that term is going to go away. Now let's continue to simplify, we're gonna multiply by two. So we get that the escape speed squared is going to be equal to two multiplied by big G multiplied by the mass of the sun divided by R one. What you can see is that when we divided out that mass of the spacecraft, what's really happening there is we're saying that the mass of the spacecraft doesn't matter. The escape speed does not depend on that mass, which was neat. All right, one final thing to isolate this escape speed, we are going to take the square root. And so escape speed is going to be the square root of two capital G mass of the sun divided by R. And I'm gonna change this to RS. OK. We're dealing with R one which is the distance between the sun and the spacecraft in this state that we're in. And we're told that that's gonna be this value RS that we're given in the problem. So we're gonna change that to RS. And now we just need to substitute in our values. And so our escape speed V escape is going to be the square root of two multiplied my big G gravitational 6.67 multiplied by 10 to the exponent negative 11 meters cute per kilogram second scored multiplied by the mass of the sun. And again, we can look this up in a table in our textbook that is gonna be 1.99 times 10 to the exponent 30 kg. And we're dividing all of this by that distance that we were told in the problem. Rs of 69.9 multiplied by 10 to the exponent 6 m. All right. Now, in terms of units, we have kilograms and then we have divided by kilograms in that numerator. So the kilograms are gonna divide out or cancel out. We have meters cubed divided by meters. That's gonna give us meters squared. So we're gonna be left with meters squared per second squared inside that square root. When we take the square root, we get meters per second. Ex exactly the unit we want for our answer choice for the speed. And when we work all of this out on our calculator, we get that the escape speed is going to be about 1.949 multiplied by 10 to the exponent 6 m per second. That is that final answer we were looking for. So that's the minimum speed that this spacecraft needs to escape the gravitational pull of the sun. If we compare this to our answer choices, we can see that this corresponds with answer choice. B thanks everyone for watching. I hope this video helped see you in the next one.

A space station orbits the sun at the same distance as the earth but on the opposite side of the sun. A small probe is fired away from the station. What minimum speed does the probe need to escape the solar system?

Key Concepts

Escape Velocity

Gravitational Force

Orbital Mechanics

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