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$\bullet$ Volcanoes on Io. Io, a satellite of Jupiter, is the mostvolcanically active moon or planet in the solar system. It hasvolcanoes that send plumesof matter over 500 $\mathrm{km}$ high(see the accompanying fig-ure). Due to the satellite'ssmall mass, the accelerationdue to gravity on Io is only$1.81 \mathrm{m} / \mathrm{s}^{2},$ and $\mathrm{Io}$ has noappreciable atmosphere. As-sume that there is no varia-tion in gravity over thedistance traveled. (a) Whatmust be the speed of material just as it leaves the volcano toreach an altitude of 500 $\mathrm{km} ?$ (b) If the gravitational potentialenergy is zero at the surface, what is the potential energy for a25 kg fragment at its maximum height on Io? How much wouldthis gravitational potential energy be if it were at the sameheight above earth?

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a) $=1345.36 \mathrm{m} / \mathrm{s}$b) $1.2 \times 10^{8} \mathrm{J}$

Physics 101 Mechanics

Chapter 7

Work and Energ

Physics Basics

Applying Newton's Laws

Kinetic Energy

Potential Energy

Energy Conservation

Simon Fraser University

University of Winnipeg

McMaster University

Lectures

03:47

In physics, the kinetic energy of an object is the energy which it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes. The same amount of work is done by the body in decelerating from its current speed to a state of rest. The kinetic energy of a rotating object is the sum of the kinetic energies of the object's parts.

04:05

In physics, a conservative force is a force that is path-independent, meaning that the total work done along any path in the field is the same. In other words, the work is independent of the path taken. The only force considered in classical physics to be conservative is gravitation.

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a satellite of Jupiter, is…

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Io, a satellite of Jupiter…

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Volcanoes on Io. Jupiter&#…

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Predict/Calculate Volcanoe…

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Predict/Calculate An Extra…

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$\mathrm{A}$ certain volca…

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A certain volcano on earth…

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So when there's a there's a volcanic corruption, we see that the particle flies out off the ground on DH reaches, let's say a height ate before it loses all its kind of energy and then again raises background. And so that means what happens here is initially we have some Connecticut energy and that kind of candidate energy drives. All the particles are the matter go at a certain height. But since the gravitational force is acting downwards, so we are. The particles continues to lose the kinetic energy, but the potential Indy starts increasing. So when the candidate energy becomes totally zero, the particle stops. And then since ah, there's gravity constantly affecting on the matters. So it starts dropping. Righto. Let's say this is a schematic diagram of the volcanic eruption. So marries around here, and then it starts going up, and at this point, the matter loses all its kind of energy. So if it's it lost all its kind of unity. The reason is that this final point So let's call this F as in final point. And let's call this IRS in the initial point so that the final point, the velocity zero so the F zero over here, and the eye is something that we need to figure out. Also, we can say that if this is why I as in the initial this displacement on DH this why f this point as y as in the final displacement, we can say that the initial displacement is zero if the particles around here but the final displace when we can call that as AIDS or the amount the hide and the man reaches before it starts dropping. So from the energy conservation, we know that the kind of like energy at point F plus the potential energy a point if should be equal to the candidate energy point I plus the potential energy find I now at this point since the displacement is zero r, there's yeah, there's no displacement. That means we can get rid of the potential energy time on DH at the top, the final velocity zero. So we can get you know, the kind of energy time. So all we have here is you f is equal, do K I. So instead of feu f, we can just right m g times h r for our concerts y f for kinetic energy. We can write that as half and the I squared right now since we mentioned that why f is h in our case so we can substitute that So we'll have m times g times age which is equal to have them p I squared and from there we consult for V I we can see that em gets cancelled because they're comin on both sides Now the eye will be square root off two times d but applied rage. Now this sea is the gravitation force on the satellite So which is given his 1.81 meters for second scratcher will be using that. So that's 1.81 use for second squared times The hype it is ah, five into 10 to the power five meters are using that We see that the velocity is that 150 muse for second So that's the first part. In the second part, we need to figure out the potential energy at the top so you can do here is we can use u equals to m g eight equation where m is the mass of the particles which is 25 kg jeezy gravitational force off the satellite, which is 1.81 meters per second squared and ages five into 10 to the power five meters. Using that, we get the potential energy as 2.3 times tented for seven jewels. Now if is the matter would want to reach the same height on Earth than the potential energy will roughly be five times greater off that quantity and we'll see why. So let's call it you sat as new satellite and let me do the competition around here so that we can compare so for, aren't we just call it you eat. That's going to be m times G g being the gravitational acceleration on earth times the same height. So now we have 25 kg times 9.8 meters per second, squared times the height. It is five times in the par five meters and that is equal to one 0.2 times 10 to provide nudes. So yeah, we can see that this is roughly five times greater than this country. Thank you

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