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The asteroid 234 Ida has a mass of about $4.0 \times 10^{16} \mathrm{kg}$ and an average radius of about 16 $\mathrm{km}$ (it's not spherical, but you can assume it is). (a) Calculate the acceleration of gravity on 234 Ida. (b) What would an astronaut whose earth weight is 650 $\mathrm{N}$ weigh on 234 lda? (c) If you dropped a rock from a height of 1.0 $\mathrm{m}$ on 234 Ida, how long would it take to reach the ground? (d) If you can jump 60 $\mathrm{cm}$ straight up on earth, how high could you jump on 234 Ida? (Assume the asteroid's gravity doesn't weaken significantly over the distance of your jump.)

3.71 $\mathrm{m} / \mathrm{s}^{2}$

Physics 101 Mechanics

Chapter 6

Circular Motion and Gravitatio

Physics Basics

Motion Along a Straight Line

Motion in 2d or 3d

Newton's Laws of Motion

Applying Newton's Laws

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Okay. So in this problem, to find out the gravitational acceleration, we can use Equation 657 Although this equation has been used to find out the gravity of gravitational force for the Earth. But we can easily use it for their short as well. Now we are given the mass of the shirt and the radius and we can use those values and then we can solve for the gravity's an expiration noise that thiss capital Ari and small Ari are the same, so we can actually use the interchange. Now G eight will be the gravitational constant which is 6.67 times 10 to the power minus 11 Newton. We described Killer Graham squared. Then we have the mass which is for times 10 to the far 16 kg over our, which is 1.6 times 10 to the party my Denver for meters squared. And from there we get the ax elation as 0.0 10 for Teo meters per second squared, which if we convert that sending Miller it is roughly equal to one centimeter per second script. Right? So that's what the gravity are gravitation. Accession will be on that stride. Now, in the second part, we need to find out the weight off. The astronauts are weight off the object that like And if we have any object on, we know the man best. We can easily use masters gravity tto find out the way. So, in our case, the object is the astronaut. Um so, yeah, we'll be using M times G, where m is the weight off the national. Now we're not given the weight. We're not given the mass of the astronaut directly, but were given the weight of the astronaut on art so we can use this equation to find out its final his mast. So from here, Mass is wait on Earth by gravitational acceleration and we can use this number over here to find out the weight on the asteroid. So weed is W e. So W is given a 6 50 Newton over G ive. This is 9.8 meters per seconds squared. Then we had the tradition acceleration which is on an asteroid with zero point do 10 for two meters per second squared. And from here we get the weight as 0.6 names. So it would be really light in this air strike. Okay, So for part C, we need to figure out the time. And ah were given eso. Basically, what we're doing is we are dropping Iraq from one meter. So in our case, Delta X is one meter. Now we need to be careful about the sign here because if we consider upward direction as our positive or taxes, then anything that's going down or has a direction downwards with B negative. So since the rocket's going downwards so our Delta X will be negative, which is one point negative one 0.0 meter and the initial velocity zero, which is V. O X and in Donkey's that serial because it's, ah, falling from a steady state. And then the acceleration will be negative off Jia's well, because its downward on direction. And if we use these conditions, we can solve for the time. So we can easily get rid off this part because the zero then so if we saw it for time here, will get twice Delta X over X. And then there's a square on in our case exes G et so we can use that. So So let's use that in the next line. So we see that Delta axis negative. And it is negative as wells on DH since Del Texas and in Britain, Exes into nominally we can get rid of this native science because they will cancel each other. So all we have is ah, two times tell takes now. Okay, let's actually put the numbers and then get rid of the signs. Um, that's negative to applying to Syria one. So you're for two meters per second squared now as we see here that there's a negative and there's a native on top so we can get rid of these two science. So we end up having the time as 14 seconds. Okay. And so, for the final part of what we need to do is we need to come here. Ah, what happens when ah, what happens? Ah, when national jumps Now we can use the question to 0.13 and we see that when astronaut is jumping, the final velocity is zero because at maximum height there's no velocity there, So final velocities zero there and ah, then if we actually put Delta X on one side and take the ratio on art versus the asteroid will end up having the following question. So for asteroid, let's call it 10 Dykes, eh? That will be so let's get rid off this part first because this is you know, so yeah, then takes a will. Be negative v o X squared. Bye. Place off, XO. In our case, X is nothing but the arts gravity. So Ah, asteroid's gravity. So that's G s. Sorry. And ah, then for the earth, it's then takes e which is equal to a native v o X and ah, divided by twice off G E. Right now, if we take this year off, uh, these two things we'll have 10 tax and over dental X e is equal to negative b oh, ex by two gs divided by negative b o x by t and we can get it off this quantity now. So we have and also this quantity as well and the negative science. So we I have now is do you over G right now we know that How much do you know that in our Thomas distance the answer traveled so we can use that to find out the distance that he will travel when he's on the asteroid. So this is T E over tea. A times Delta X Right on. We know that G e is 9.8 meters per second squared and then G is 0.1 zero for two. We never second squared and then Del the exit is 0.6 year o meter and against us, the distance off 5.6 times 10 to the part two meter R 5 60 meters. Now, as we see that, it's ah really, really high jump. So if the distance is really hide, then one thing that we should note here is that the gravity will change if the height is really if, ah, the hide a significant but and we can actually see that from the equation which gives us the relation between gravity and the distance. So if we actually see this equation over here, we have detained and over r squared. So we see that gravity causes inverse off r squared. So that means if are significantly high, our gravity will be reduced. And in turn, we see that in this case, when the asteroid is something 5 60 meters up, then the gravity's changing by 6%. So there's a drop off 6% off the gravity ofthe the origin of gravity or the gravity when it's on the on the surface on the asteroid is on the surface. But we we are not taking this into consideration because just to simplify the problem, we were just taking this gravity into account. And that gives us 5 60 meters. Thank you.

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