Ten days after it was launched toward Mars in December 1998, the Mars Climate Orbiter spacecraft (mass 629 kg) was 2.87 * 10^6 km from the earth and traveling at 1.20 * 10^4 km/h relative to the earth. At this time, what were (a) the spacecraft's kinetic energy relative to the earth and (b) the potential energy of the earth–spacecraft system?

Video transcript

Welcome back everybody. We are taking look at a rocket that is returning from its mission to Mars. It's just a little bit smaller because we need to work with the room between here. Now it's returning from Mars going to earth and we are told a couple of different things about this trip. We're told that the mass of the rocket is 750 kg. Were told that the mass of Mars is 6.4, 2 times 10 to the 23rd kg. We are told here that the radius of Mars is 3390 km. And we are looking at a specific point in this journey. Specifically when the rocket is at a distance from the surface of Mars of 10 to the 5km. We are told that at this distance it has reached a cruising speed Of 36, km/h. We are tasked with finding two things. This is a lot of information, but it will get simpler. I promise. You're tasked with finding two things. One, the kinetic energy of the rocket relative to Mars. And then we are also tasked with finding the rocket Mars potential energy. Let's go ahead and start with our kinetic energy. The formula for kinetic energy is simply this, one half times the mass of our rocket times the velocity squared. So let's go ahead and plug in our values here. If one half times 750 kg times 36,000 kilometers per one hour. But we need this in meters per second. So I'm gonna convert real quick in one kilometer. There is 1000 m and in one hour there is 3600 seconds. All of these terms are going to cancel out leaving us in our meters per second just for the velocity here. But What this results in when you plug it into our calculator is 3.75 times. And to the duels for our kinetic energy. Great. Moving on to part B here, we want the gravitational potential energy of the system here. So this is going to be equal to the negative of Newton's gravitational constant times the mass of object one, which we'll just say is mom times the massive object to which is the mass of the rocket. All over the total distance between the rocket and the center of mars, which as you can see is going to be our plus great as before. Let's just go ahead and plug in all the values since we know it's going to be negative 6.673 times 10 to the negative 11. That's Newton's constant times the mass of MArs which is 6.42 times 10 to the 23rd Nice master rocket Which is all over radius of MArs which is 3390 plus the distance between the rocket and the surface of mars which is 10 to the fifth kilometers. But we need all this in meters. So we will multiply all of that by 10 to the third, plugging this into our calculator, we get that are gravitational potential energy of the system is negative 3.1 times 10 to the eighth jewels. Now we have found the kinetic energy and the potential energy building us an answer of D. Thank you all so much for watching hope this video helped. We will see you all in the next one.