The Parker Solar Probe, launched in 2018, was the first spacecraf...

Question

The Parker Solar Probe, launched in 2018, was the first spacecraft to explore the solar corona, the hot gases and flares that extend outward from the solar surface. The probe is in a highly elliptical orbit that, using the gravity of Venus, will be nudged ever closer to the sun until, in 2025, it reaches a closest approach of 6.9 million kilometers from the center of the sun. Its maximum speed as it whips through the corona will be 192 km/s.(b) The probe's highly elliptical orbit carries it out to a maximum distance of 160 Rₛ with a period of 88 days. What is its slowest speed, in km/s?

Accepted Answer

Hey, everyone in this probably we have a space agency that's planning to send spacecraft to explore Venus. Hey, it wants to put the spacecraft into a 24 hour elliptical orbit told that the spacecraft will reach an altitude of 300 kilometers from Venus when it's at its closest distance to a planet and 700 kilometers. When it is at its furthest distance, the spacecraft's maximum speed at its closest distance is 400 kilometers per hour. And we want to calculate the spacecraft speed, ok. At its furthest distance from Venus, you have four answer choices all in kilometers per hour. Option, a 171 option B 320 option C 491 and option D 933. All right. So when we have some distances here, we have information about speed, we want to find information about speed. So let's think about our angular momentum kind of contains all of those values. OK. So we have no network acting on this spacecraft and we're gonna assume that there's only the gravitational force acting on it. And so angular momentum is conserved if angular momentum is conserved. We can write the L one is equal to L two. OK. The angular momentum at state one is equal to the angular momentum in state two. Now, what is angular momentum or recall that we can write angular momentum as the cross product between the distance or displacement vector and the linear momentum. OK. So we can write this as R one cross P one is equal to R two cross P two. And recall that we can simplify that further and write this as R one M V 19 theta one is equal to R two M V two data two. Hm. All right. So we have R one and R two, which is gonna be the distance from our reference points. So in this case, it's gonna be the distance from Venus, we have the mass M which we can divide on both sides because that space we have just that spacecraft. So the mass when the spacecraft is close and the mass when the spacecraft is far is the same, we have V one and V two. So those are those speeds. So the closer and the further speed respectively and then we have sine beta one and sine beta two. And those data values are gonna be the angle between the position vector and the velocity vector at the given point. OK. So let's think about the location of the closest and furthest points. OK. Those are gonna lie on the major axis in this elliptical orbit. OK. We're talking an elliptical orbit here. So if we were to draw our elliptical orbit, something like this Venus is at this black dot Our closest point is gonna be on the left hand side at the major axis there. Hm R one is going to point from that spacecraft inwards to Venus and the velocity vector V one is gonna be pointing upwards and at this point, the velocity is completely pointing upwards. And so we have V one, you can see that the angle between V one and R one which we call theta one is just gonna be 90 degrees. Now, on the other side, and this is gonna be our furthest distance. So on the major axis on the far right, that's gonna be our furthest distance. OK. And so we have R two that's gonna point to the center and I'm not gonna draw it all the way across, but it's gonna point all the way to Venus. And we have the velocity is gonna be going straight downwards at this point. OK. So again, the angle between them, it's gonna be 90 degrees. All right. So now we can write the R one multiplied by V one multiplied by sine of 90 degrees is equal to R two multiplied by V two multiplied by sine of 90 degrees. A sin of 90 degrees is just one. So both of those terms go to one or you could think about dividing the entire equation by sine of 90 degrees if you want to think about it that way. And what that tells us is that V two, the velocity or the speed at the furthest distance, we can write as R one, V one divided by R two. OK. So the speed at the furthest distance is going to be related to the speed at the closest distance with some ratio of their distances. Now, we know R one, we know V one and we know R two. So at this point, we just need to substitute in these values and solve. OK. The hard part is done, but let's go ahead and do that. R one at closer distance is 300 kilometers. V one, the speed of the closest distance is 400 kilometers per hour. And we divide that by that further distance of kilometers, the unit of kilometers will divide out. We're gonna be left with kilometers per hour, which is what we want for that speed. We get that this is going to be equal to 0.42, 86 kilometers per hour. All right. So we found the speed of the spacecraft at that further distance. And one thing to note here is we divided out that mass. So it didn't matter what the mass of the spacecraft was. All that mattered was the instances at the closer and further points as well as the speed at that closer point. A if we round to the nearest kilometer per hour to match the answer choices, we can see that this corresponds with answer choice. A 171 kilometers per hour. That's it for this one. Thanks everyone for watching. I hope this video helped.

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