Two satellites are in circular orbits around a planet that has ra...

Question

Two satellites are in circular orbits around a planet that has radius 9.00 * 10^6 m. One satellite has mass 68.0 kg, orbital radius 7.00 * 10^7 m, and orbital speed 4800 m/s. The second satellite has mass 84.0 kg and orbital radius 3.00 * 10^7 m. What is the orbital speed of this second satellite?

Accepted Answer

Welcome back everybody. We are looking to launch satellites around some unknown planet right now. We already have one existing satellite, which I'm just going to label with s orbiting around. And we are told a couple different things. We are told that the mass of our existing satellite is 240 kg. Were told that its orbital speed is 9449 m per second. And we are told that its radius of orbit is 6.5 times 10 to the eight m. Now, we want to launch a brand new satellite. Right? So a new satellite, which for new, I'm just going to label with N. And we are told that the mass, This new satellite, it's 305 kg and that its radius of orbit is 2.05 times 10 to the 8th m. And we need to figure out what the orbital speed is that we should provide to our new satellite. Here's what we're gonna do. I'm gonna use I'm gonna use some some laws here, Right. One of these laws states that are gravitational force of of the bodies of orbit around our unknown planet here is equal to Newton's gravitational constant times. Let's see here, the mass of our big planet, which we don't know what that masses times the mass of an individual satellite. All over the radius of orbit squared. So let's just pull this equation aside real quick. This equation, we can actually simplify a little bit further, Right? We like I said, we don't know what this mass is. So we need to fill in for that. Well, we know that when dealing with orbits, we know that orbital speed is equal to the square root of Newton's gravitational constant times the mass of our planet. In this case times the radius of our planet. Right? So let's see here, let's see. We have, if we square both sides, we have that V squared is equal to big G times M over R. So let's plug this into this equation. We get that F is equal to G. M times little M. All over R squared, which is equal you M B squared over R multiplying by R squared on both sides and divided by little M on both sides. We have that G M is equal to r V squared times some constant. And you'll notice that G M G just being Newton's gravitational constant and envying the mass of this planet is also going to be some constant. So we can set up some equations from this as well for the existing satellite, then we have that its radius of orbit times its velocity squared is equal to some constant. And then we have uh for the new satellite we have that its radiance of orbits times its velocity, which is what we're trying to find, is equal to some constant as well. So we're gonna set these two equations equal to one another. So let's go ahead and do that down here, we have that. Our N V and squared is equal to R. S. V S squared. Which when solving for V. N, we get that V N is equal to V. S times the square root of R. S over R N. We know those values. So let's go ahead and plug those values in. We have that the velocity of our existing satellite is 9449 times the square root of the radius of our existing satellite of 6.5 times to the eighth. All over the desired radius of orbit of our new satellite, 2.5 times 10 to the eighth. Giving us a desired new orbital speed of our satellite of six, 18,800 m per second corresponding to our answer choice of E. Thank you all so much for watching. Hope this video helped. We will see you all in the next one.

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