You have been visiting a distant planet. Your measurements have determined that the planet's mass is twice that of earth but the free-fall acceleration at the surface is only one-fourth as large. (b) To get back to earth, you need to escape the planet. What minimum speed does your rocket need?

Video transcript

Hey, everyone in this problem, we have a spaceship landing on an exoplanet and the conditions end up being extremely dangerous. So the crew decides to escape the exoplanet as soon as possible. They using the spaceship instruments only, they measure the gravitational acceleration of the exoplanet to be hacked out of earth and its mass to be a third of that of earth. And we're asked to calculate the minimum speed necessary to escape the planet's gravity. We have four answer choices all in kilometers per second. Option. A 4.8 option B 7.2 option C 11.2 and option D 12.4. So we're interested in this escape velocity for the spaceship and let's consider the conservation of energy. OK. So the conservation of energy tells us the mechanical energy is going to be conserved and the mechanical energy is the sum of the kinetic energy and the potential energy. So we have K one plus the U one, it's going to be equal to K two plus the U two KK is the kinetic energy U is the potential energy. We're gonna use the subscript one to indicate that we're at the surface of the exoplanet. And the subscript two is going to be one more added distance R from the planet. Capital R. OK. So our kinetic energy recall is given by one half MV squared. We're gonna use that both sides of this equation. And then we have our gravitational potential energy. It is U is potential energy. We don't have to worry about any springs here. The only potential energy we have is gravitational potential energy. And when we're talking about being a distance from a planet, we're gonna have to talk about the gravitational potential energy given by G. Yeah, M divided by R. And that stems from Newton's law of gravitation that we end up with that gravitational potential of energy. So let's go ahead and write this all out. Yeah, we're gonna have one half MS V one square. OK. And FS is gonna indicate the mass of our spa. OK. So that's our kinetic energy. The gravitational potential energy is negative. OK. So we have negative capital G, the gravitational constant multiplied by me, which is gonna be the mass of the exoplanet multiplied by the mass MS of the spaceship all divided by the radius re of the Oplan. OK. So that's the left hand side of our equation. On the right hand side, similarly, we get one half MS V two squared mind G me MS divided by R and remember that R is the distance that we are from the planet all right. So to calculate escape speed, recall that we have to take two conditions, we have to consider that the velocity V two going to 0 m per second, see the velocity as we get further and further and further away from the planet. And we have to consider that our distance R is going off to infinity. And so as we get further and further and further away from this planet, our speed is going down to zero. Now, if we take these two conditions and we substitute them into our equation or apply them to our equation, we're gonna find that the entire right hand side goes to zero. OK? Because V two is going to zero. So that deals with the first term and then R which is in the denominator is going to infinity. So that term is gonna go to zero as well. And what we're left with then is that one half the max MS multiplied by our escape speed VESC squared minus G me MS divided by re is equal to zero. And we've just replaced V one with V escape just to be clear that we are talking about our escape speed there. So now you have in this equation and we can solve for our skate speed. First thing we're gonna do is move our entire gravitational potential energy to the right hand side. So we can write one half MS B escape squared is equal to G me, MS divided by art. All right. Now, let's simplify as much as we can. This mass of the spaceship MS, we can divide both sides by that. We're gonna multiply both sides by two and then take the square root and we get that the escape speed, the escape is gonna be the square root of two GME divided by re All right. Now, we have our escape speed in terms of the mass of the exoplanet, the radius of the exoplanet. Remember we aren't given those values in terms of numbers, we're giving them in terms of their ratio to that earth and that of, yeah, to that of earth. All right. So what substitute in the relationship we're given, we have that the escape speed is going to be equal to the square root two G multiplied by the mass of the exoplanet, which we're told is one third, the mass of the earth. OK. So we have two G multiplied by one third mass of the earth dividing by the radius of the exoplanet, the radius of the exoplanet related. OK. We're told the relationship between the mass and we're also told the relationship about the gravitational acceleration of the exoplanet and the earth. OK. So let's go off to the side and see how we can calculate the radius of the exoplanet. All right. Now, we know that the gravitational acceleration on the exoplanet GE is one half of the gravitational acceleration on earth GE. And again, we know that the mass is one third. Now let's go ahead and substitute what we know about little G on earth. Little G, the gravitational acceleration on the exoplanet, we can write as negative G multiplied by the mass of the exoplanet divided by the radius of the exoplanet square. Hm. Right. And we know that that is equal to G divided by two, which is negative G multiplied by the mass of the earth divided by two times the radius of the earth squared. So we're gonna work with the right part of this equation, the negative G, we can divide from both sides. OK. And we want to find an expression for the radius of the exoplanet re, what we have right now is that the mass of the exoplanet divided by the radius of the exoplanet squared is equal to the mass of the earth divided by two times the radius of the earth. Where, and so we can write that the radius of the exoplanet is squared is equal to the mass of the earth multiplied by two times the radius of the earth squared divided by and sorry, acid mass of the earth and the mass of the exoplanet. OK. And then we're dividing by the mass of the earth. All right. So we're getting closer, we have an expression for RE squared. We want an expression for re but let's remember what we're told about the mass of the exoplanet. OK? Because we can simplify that as well. And we don't want this me in our equation because we don't have a numerical value for it. OK. So we want to get this in terms of the mass of the earth, the radius of the earth, those values that we know. So we can write that the radius of the earth squared is equal to the mass of the earth divided by three multiplied by two times the radius of the earth squared divided by the mass of the earth. OK. We're replacing the mass of the op with the mass of the earth divided by three, like we were told in the problem. And then we can see that that mass of earth is going to divide it. We're gonna be left with the radius of the exoplanet being equal to the square root of two thirds multiplied by the radius of earth. All right. So that was a little bit messy on the side. But what we use is we just used the information we were given about the relationship between the gravitational acceleration on the exoplanet and on earth to write down a relationship for the radius of the exoplanet in terms of some values that we know. OK. So if we go back into our escape velocity or our escape speed equation, we can now write re the radius of the exoplanet in the denominator as the square root of two thirds multiplied by the radius of earth. All right. We can simplify before we substitute in some numbers. So we end up with this escape speed is the square root. We have two thirds and then we're dividing by the square root of two thirds, which is the same as multiplying by the square root of three halves. And then we have g multiplied by the mass of the earth divided by the radius of earth. Let me give some more room here and we are gonna substitute all these values in now and calculate this escape speed. So we get the escape speed is going to be the square root of what? Well, it's gonna be the square root of the square root of two thirds multiplied by 6.67 times 10 to the experiment negative meters cubed kilogram second squared multiplied by 5. times 10 to the exponent 24 kilograms. OK. The mass of the earth, you just extend that square root and we're dividing all of this by the radius of the earth which is 6. times 10 to the exponent 6 m. All right. So it looks messy, but we can plug all of this into our calculator in terms of the unit, the unit of kilogram is gonna divide it in the numerator. We have meters cube divided by meters. And so we're gonna be left with meters squared per second squared. Let me take the square root, we get meters per second, which is what we want for a speed. And we get that the escape speed is about 7150. m per second. Remember that the answer choices were given in kilometers per second. So we want to convert to kilometers divided by 1000. We're gonna round to two decimal places and we get that the escape speed is about 7.15 kilometers per second. If we go up to our answer traces and compare what we found and we're gonna round a little bit more to one decimal place. We can see that. The correct answer here is option C seven or sorry. Option B 7.2 kilometers per second. Thanks everyone for watching. I hope this video helped you in the next one.