A 4000 kg lunar lander is in orbit 50 km above the surface of the moon. It needs to move out to a 300-km-high orbit in order to link up with the mother ship that will take the astronauts home. How much work must the thrusters do?

Video transcript

Hello, fellow physicists today, we're gonna solve the following practice problem together. So first off, let's read the problem and highlight all the key pieces of information that we need to use. In order to solve this problem. A spacecraft with a mass of kg is in a circular orbit kilometers above Jupiter's surface. The spacecraft must use its thrusters to transition to a higher circular orbit kilometers above the surface. Given that Jupiter has a radius of 69,911 kilometers and a mass of 1.89 multiplied by 10 to the power of 27 kg. Calculate the amount of work required for the thrusters to move the spacecraft from its initial orbit to the higher orbit. So that's our end goal is to calculate the amount of work required to the required for the thrusters to move the spacecraft from its initial orbit to the higher orbit. So from 200 kilometers to 600 kilometers, awesome. So we're given some multiple choice answers and they're all in the same units of jewels. So let's read them off to see what our final answer might be A, is 1.3 multiplied by 10 to the power of eight. B is 2.3 multiplied by 10 to the power of 10 C is 1. multiplied by 10 to the power of 13 and D is 2.1 multiplied by 10 to the power of 15. So first off, let us recall and use the equation for velocity that describes a spacecraft of mass M in a circular orbit. So that would state that velocity squared is equal to the gravitational constant multiplied by the mass of the planet divided by R where R is the distance from the center of the planet to the spacecraft. Thus, when we take into account, the kinetic energy, we can state that K is equal to or the kinetic energy is equal to one half the mass of the spaceship lowercase M is capital M is the mass of the planet multiplied by the velocity squared is equal to the gravitational constant multiplied by the mass of the planet multiplied by the mass of the spaceship divided by two multiplied by R which is the distance from the center of the planet to the spacecraft. Awesome. So note that the gravitational energy let's call it U subscript capital G. So U subscript capital G, the gravitational energy is equal to negative gravitational constant multiplied by the mass of the planet multiplied by the mass of the spaceship all divided by R which is the distance between the center of the planet to the spacecraft. So we can go on to say that K is equal to negative one half U multiply or capital U multiplied by G. So negative one half multiplied by capital U multiplied by capital G. Let's also note that the mechanical energy E subscript mem is equal to U G plus K which is equal to one half multiplied by UG. So now we can solve for the amount of work required to move the spacecraft from 200 kilometers to 600 kilometers will be that the work W is equal to delta em is equal to one half, delta UG which is equal to one half multiplied by naked G, the gravitational constant multiplied by the mass of the spaceship multiplied by the, actually the mass of the planet capital M multiplied by the mass of the spaceship lowercase M divided by R H which is the higher altitude or higher orbit value, which would be the are actually the by the higher. So the higher, yeah, higher over it minus G multiplied by the mass of the planet multiplied by the mass of the spaceship divided by R lower RL. OK. So now we can simplify this equation to W the work is equal to one half multiplied by the gravitational constant multiplied by the mass of the planet multiplied by the mass of the spaceship, all multiplied by one divided by the lower which the lower one would be 200 kilometers and the higher one would be kilometers just to be clear, I are lower, minus are higher and keep it straight in our mind. So now we can plug in our known variables. And so, so let's do that. So the work is equal to one half multiplied by the gravitational val gravitational constant. And the numerical value for that is 6.67 multiplied by 10 to the power of negative 11. And its units are newtons multiplied by meters squared, divided by kilograms squared multiplied by the mass of the planet which is 1. multiplied by 10 to the power of 27 kg multiplied by the mass of the spaceship which was kg multiplied by. And let's move this down since we're running out of room here, one divided by six 69 plus 200 which we need to convert this from kilometers to meters. So we could you just multiply all this by 10 to the power of 3 m minus one divided by 69,911 plus 600 multiplied by 10 to the power of 3 m. So when we plug that into a calculator, we should get that. The work done is 2.3 multiplied by 10 to the power of 10 jewels, which is our final answer. We did it folks. So that means our final answer. Looking at our multiple choice answers has to be the letter B 2.3 multiplied by 10 to the power of 10 jewels. Thank you so much for watching. Hopefully that helped and I can't wait to see you in the next video. Bye.