A satellite follows the elliptical orbit shown in FIGURE P12.77. The only force on the satellite is the gravitational attraction of the planet. The satellite's speed at point 1 is 8000 m/s. b. What is the satellite's speed at point 2?

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, an exoplanet moves in an elliptical orbit around its sun. As depicted in the diagram. The two planets are attracted to each other by a gravitational force, the speed of the planet at the vertex V subscript one and of its trae trajectory is 800 kilometers per hour. Calculate the speed of the planet at the second vertex V two V subscript two. OK. So that's our end goal is to calculate the speed of the planet at the second vertex. Awesome. So we're given some multiple choice answers. They're all in the same units of kilometers per hour. So let's read them off to see what our final answer might be. A is 400. B is 480 C is 800 D is 1000. OK. So here as the prom state, we have a diagram here to the right down below it shows us that D one is equal to two multiplied by 10 to the power of eight kilometers. D two is eight multiplied by 10 to the power of eight kilometers. And D three is equal to 1.8 multiplied by 10 to the power of eight kilometers. And it shows it with arrows for D one is in blue and D two is in green and D three is in red. It also shows the first vertex and the second vertex. And it also shows the planet in the sun. And the elliptical orbit is the black circle and it has our Y axis and X axis or AY. Perfect. OK. So first off, let us recall that the net torque applied to a planet is zero. Therefore, the angular momentum of the planet is conserved. So we can write that delta vector L is equal to zero vector and that vector in that vector L vertex one is equal to vector L vertex two. Awesome. So let's write our first important equation M multiplied by vector R one cross multiplied by vector V, one is equal to M multiplied by vector R two cross multiplied by vector V two. OK. So note that the, that at vertex one and vertex two, the velocity vector is perpendicular to the vector R which is the distance from the sun to the planet. Also note that at vertex one, the distance separating the planet from the sun is D two minus D one. OK. So let's make a quick note of that. So at V one, D two is minus D one K. And at vertex two, the distance separating the planet from the sun is D two plus D one. OK. So when we apply the information above to equation one, we will get that D two minus D one multiplied by V one is equal to D two plus D one multiplied by V two. So we need to rearrange this equation to solve for V two, the second vertex using some algebra. So let's do that. So V two is equal to D two minus D one multiplied by V one divided by D two plus D one. So now at this stage, we need to plug in all of our known variables and solve for the numerical value for V two. So V two is equal to D two. So D two was eight, multiplied by 10 to the power of eight kilometers minus two, multiplied by 10 to the power of eight kilometers divided by eight, multiplied by 10 to the power of eight kilometers plus two, multiplied by 10 to the power of eight kilometers. OK. And that is all multiplied by 800 kilometers per hour, which I ran out of room here. So it's all multiplied by 800 kilometers per hour. So when we plug that into a calculator, we should get B two is equal to 480 kilometers per hour. So the 800 kilometers per hour, in case you're confused where that came from. That's the value for V one. Ok. So we did it. We found the second factor, the speed of the second vertex, I should say not vector is 480 kilometers per hour. Ok. So let's look at our multiple choice answers to see what the correct answer is. The correct answer is the letter B 480 kilometers per hour. Thank you so much for watching. Hopefully that helped and I can't wait to see you in the next video. Bye.