The 75,000 kg space shuttle used to fly in a 250-km-high circular orbit. It needed to reach a 610-km-high circular orbit to service the Hubble Space Telescope. How much energy was required to boost it to the new orbit?

Video transcript

Hi, everyone. Let's take a look at this practice problem dealing with circular orbits. So in this problem, a satellite of mass 250,000 kg is in a circular orbit of height of 550 kilometers above the surface of the earth needs to move to an orbit that's at a height of 1500 kilometers from the surface of the earth. How much energy will be required for this transition? Given four possible choice as our answers voice A is negative 2.7 multiplied by 10 to 13 joules. Choice B is 1.7 multiplied by 10 to the 12 joules C is 8.7 multiplied by 10 to the 11 joules. And choice D is 2.7 multiplied by 10 to the 13 Js. Now, since we're moving from one circular orbit to another, we do have a formula or the amount of energy that is required for this transition. To recall that the energy required for the transition E is equal to one half delta U. So it's equal to one half our change in our potential energy. And here our potential energy is our gravitational potential energy recall a formula for the gravitational potential energy we have U is equal to minus G little M multiplied by capital M divided by R where G here is the gravitational constant, the little M is gonna be the mass of the satellite capital M is the mass of the planet and R is the radius of the orbit. So we can plug this um expression for potential energy into our um energy formula. So we have E is equal to one half multiplied by. And I'm gonna have two terms here for our use our initial and final and each term is gonna have the same um G little M and capital M. So I'm gonna factor those out. So I have a negative G little M multiplied by capital M. And this is gonna multiply the quantity of one divided by R two minus one divided by R one. Here are two. It's gonna be our final orval radius and our one is gonna be our initial orbital radius. Now, if I look at what I was given the problem, I was actually given the height. So I was given the altitude above the surface, not the radius of the orbit, I need to calculate the two different radii. So we'll start with R one, this is gonna be my initial um or radius. So our one is gonna be equal to I have my initial height which is gonna be the 550 kilometers and I'll need to add to that the radius of the earth. I'll have a plus 6371 kilometers. When I add those together, I get R one is equal to 6921 kilometers. However, I noticed that the final answer that I'm gonna be looking for, it needs to be in jewels and for that to occur, I'll need to convert my kilometers into meters and I can do that just by multiplying by 10 to the three. So I have R one is equal to 6921 multiplied by 10 to the 3 m. So I'm gonna do the same thing with R two, I have R two. It's gonna be equal to and for its altitude, that was the 1500 kilometers they'll have plus the ready to earth, which is 6371 kilometers. So it gives me value for R two of 7871 kilometers. And again, I'll need to convert that to meters by multiplying by 10 to the three. We'll have R two is equal to 7871 multiplied by 10 to the 3 m. So now that I've calculated my orbital radii, I can now plug those into my formula for the um energy that's required for this transition. Coming back to our formula for the energy we have E is equal to, I'll have a negative one half and four G, I'll have 6.67 multiplied by 10 to the negative 11 Newton meter squared per kilogram squared. They'll need to multiply that by the mass of the satellite, which is the 250,000 kg. And I'll need to multiply that by the mass of the earth which we can look up. And that is 5.97 multiplied by 10 of the 24 kg. And this needs to be multiplied by the quantity of one divided by R two, which is the 7871 multiplied by 10 to the 3 m. Then I'll have minus one divided by R one which is the 6921 multiplied by 10 to the 3 m. So now that I have um just a value on the right hand side of that equation, I can now calculate my value for the energy. And so I have E as equal to 8.7 multiplied by 10th, the 11 rules. Here, I just kept two significant figures. And when I look at what answer choice that corresponds to that corresponds to answer C. So the trick to this problem was actually being able to identify the different orbital radii. And so here we, you had to recognize that you were given an altitude or height above the earth's surface and not the radius of the orbit. So you need to add the radius of earth to each of those um heights. So I hope that this has been useful and I'll see you in the next video.