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In July 2005, $NASA$'s "Deep Impact" mission crashed a 372-kg probe directly onto the surface of the comet Tempel 1, hitting the surface at 37,000 km/h. The original speed of the comet at that time was about 40,000 km/h, and its mass was estimated to be in the range (0.10 - 2.5) $\times$10$^{14}$ kg. Use the smallest value of the estimated mass. (a) What change in the comet's velocity did this collision produce? Would this change be noticeable? (b) Suppose this comet were to hit the earth and fuse with it. By how much would it change our planet's velocity? Would this change be noticeable? (The mass of the earth is 5.97 $\times$ 10$^{24}$ kg.)

A. $1.4 \times 10^{-6} \mathrm{km} / \mathrm{h} .$ This change is not noticeable.B. $6.7 \times 10^{8} \mathrm{km} / \mathrm{h} .$ This change is not noticeable.

Physics 101 Mechanics

Chapter 8

Momentum, Impulse, and Collisions

Section 3

Momentum Conservation and Collisions

Moment, Impulse, and Collisions

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problem. 8.35. So we're told about this mission that NASA did where they ran a probe shown here in red into a comet shown here in blue and vastly not to scale. We're told the masses of these things. The probe wave had a massive. He added 72 kilograms and the comet with this sort of thing, the only contest emate the mass and were told to use the lower end of the estimate, which is 0.1 times 10 to the fourth and the probe as it approached the comet. It's speed right before impact. Was this 37,000 kilometres per hour. With respect to become so we want to find out is what change in speed of the comet is this produce? And if the comet were to crash into the earth, What speed with that cause what change in speed of the earth with that cause. And then in both cases, we want to know whether that would be signature again. And the comet is going to be moving at 40,000 miles relative to the heiress and they're its mass is also given. But we'll worry about those in part So for part A, we know that momentum. It has to be conserved. So they're no external forces acting on these. They're far out in space. You know, nothing. Nothing can act on them externally. That is going to be significant. So the initial momentum, it's going to be the mass of the probe times. We'll call it its initial speed. And this has to be equal to the mass of the two combined things moving at the final speed of the comet, which, because we're assuming starting off with the comment being at rest and did a reference frame where it's going to be using. Yeah, the final speed of the comment will also will be the same as the change in its. So this will be the mass of the probe us. Massive comment I And so then we just divide both sides by the combined mass. And I could just said the final speed of the comment because we're starting in a reference brain where were considering everything in a reference frame where the comet is arrest, its final speed will be people to change in speed, and so this is equal to the ratio of the mass of the probe to the mass of the combined system of Probe and Comet finds the initial speed of the problem and plugging numbers in the initial speed of the probe. Despite is quite high, uh, but mass of the comet is going to be what, what ends up dominating sort of value. We end up getting out here because 10 to 14 kilograms is gigantic, attended the 30. Either way, um, and so this works out to be 1.4 times and the negative six kilometers. So even though this is moving quite fast, it's it's speed. This 10 orders of 11 orders of magnitude smaller than the Ritz masses and Levin orders of magnitude smaller than this and so only about 1,000,000 of a kilometer per hour is changing speed that it produces. And even even without any without like as a relative speed to the, you know, sort of characteristic speed we have here. This is very small, probably wouldn't be able to measure this very easily. So now let's assume that the comment then hits the Earth. The Mass of the Earth. His 5.9 seven times 10 with 20 forests now notice again. Mass of the comet is about 10 of the 13 kilograms, and here we have six ish times 10 to the 24. So this is again, like a 10 10 11 order magnet order of magnitude difference that we're gonna probably expect a similar thing to occur where the speed changes going be negligible. But of course, we have actually find out if that's the case. And certainly if we want an actual number, just sort of intuitively talking about it isn't going to get us there. So nothing about the actual physical situation exchange. So we know that the change in speed of the earth I mean, basically, we just put in the e wherever see, was here and then see whoever he was here. So the change in the Earth's speed is going to be equal to a massive dick on it, plus the mass of the probe, which is negligible compared to this. But you know, for the sake of being complete, you should include that and for us, and this is the comet is going 40,000 kilometers per hour with respect to the earth. It's a plugging all of the numbers in we find that the change in the earth's speed is a whopping 6.7 times 10. It's the negative eighths kilometers, which is even quite a bit smaller than this Thea difference in terms of the order of magnitudes, of the things you're a similar, it still ends up being quite a lot less because that's not quite how it just for comparison purposes. And again, this was this was driving. If this is difficult to even measure, is it also being even more difficult to measure? So this is not significant in either case. Now, of course, uh, the actual effects of a comet this size impact and hear it is gonna be quite catastrophic, probably because it's not the change in the Earth's speed, because is the problems. It's, you know, amount of energy that gets deposited into the surface of the earth all at once. He kicks up dust and causes tidal waves and things. That tent, you know, cause problems for living things on Earth. The best of my knowledge, this comment didn't end up hitting the earth, which I think we can be grateful for, because although it wouldn't have changed how fast they're going very large, it it would have been disastrous

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