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

(a) $1.4 \times 10^{-6} \mathrm{km} / \mathrm{h}$

(b) $\Delta v=6.7\times 10^{-8} \mathrm{km} / \mathrm{h}$ , this change is not noticeable

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In this question, a probe crashed at the surface of the comet at a velocity off 37,000 kilometers per hour. The mass of that probe is 372 kg, while the mass off the comet is 0.1 times 10 to the 14 kg. Now what is the change in the velocity off the comet after that collision? For that, you have to remember about momentum conservation. So momentum, this quantity right here is always conserved. It goes as follows. The net mo mentum is always always conserved. This is a law off physics. Then using that law, we can say the following the net momentum before the collision must be equals to the net momentum after the collision. Okay, so in this question, we are working at the comments reference frame. So the comet is at rest with respect to its own reference frame. This is why we're not writing the velocity of the comet, which is very, very fast to it is 40,000 kilometers per hour. But since we are in the comments reference frame, its velocity in that reference frame is the quest to zero. So the net momentum before the collision is given on Lee by the momentum off the probe. So the momentum off the probe is given by this expression. Then we have the mass off the probe times the velocity off the probe and this must be close to the net momentum after the collision. After the collision off course both the comet on the probe. We've been traveling together. Then the net momentum after the collision is given by the following it is equals to the maths off the probe, plus the mass off the comet times the velocity off the comet on the probe now reference frame since we were working in the comments reference frame before the collision. That velocity off the set in that reference frame is exactly the variation in the velocity off the comet so we can call this velocity Delta V already. So what I'm saying is that given this reference frame, which is moving at 40,000 kilometers per hour after the collision, this set will be moving with respect to that reference frame with some velocity that relative velocity is Delta V. So the absolute velocity with respect to some other reference frame would be 40,000 kilometers per hour plus this variation in the velocity. Since we are working in a reference frame which is moving at 40,000 kilometers per hour, we Onley consider Delta V now that we have this expression we can solve for the variation in the velocity. So Delta V is because of the mass of the probe times the velocity of the probe divided by the mass of the probe, plus the mass off the comment. Now it's important to point that I'm choosing a reference frame which is moving with the same velocity as a comet in the same direction as the comet. But in my reference frame, everything that points to the right is positive. It's very important to notice that because it will matter for the sign off the velocity. The reform Delta V is given by the mass of the probe, which is 372 kg times the velocity off the probe, which is 37,000 kilometers per hour. So we have a tangle, a turn here divided by the mass of the comet, which is 0.1 times 10 to the 14 kg plus the mass off the probe 372 kg. These results in a variation in the velocity that is approximately 1.4 times 10 to minus six kilometers per hour. So the change in the velocity is very, very, very small compared with the velocity off the comet which is 40,000 kilometers per hour to reform. The change in the velocity is not noticeable. Now we go to the second part off this question in the second part off this question. That same comment is now to hit the earth at a velocity off 40,000 kilometers per hour. We already know what is the mass off the comet. The mass of the Earth, on the other hand, is 5 97 times standard 24 kg. That will be a explosive collision. Okay, we have to evaluate yet again. What is the change in the velocity off the earth after that collision on, We have to tell if that change is noticeable or not. We are again doing almost the same thing. But now, using different data and of course, a different reference frame. Now our reference frame is moving together with the earth. And let me already tell you that everything that is pointing to the right by my choice is positive, and everything that is pointing to the left as a consequence is negative. Okay, that being said, we can begin the question again. We have to use the idea off conservation of momentum. So the net momentum before the collision must be accosted. The net momentum after the collision before the collision in Earth's reference frame on Lee the comet were moving there for the momentum was given by the mass off the comet times the velocity off the comet. And then after the collision, both the Earth and the comments are moving. Now the earth is moving because there is a change in the velocity off the earth because off that collision and our reference frame moves together with the earth before the collision. So we're still moving with the same velocity as the earth. We're moving before the collision. Therefore, even our new reference frame the earth is moving after that collision. Then, after the collision, both the Earth and the comet are moving together There, for the momentum is given by the mass off the comet, plus the mass off the Earth times again, the change in velocity. The reason is that the change in the velocity is precisely the velocity that the Earth and the comet will be moving together with respect to our reference frame that is moving with the same velocity as the earth before the collision. Now that we have the equation we can solve for the VA religion, the velocity. As you can see this, this is very similar to what we have done before. So Delta V is the mass of the comet times The velocity of the comet divided by the mass of the comet, plus the mass of the earth than the variation in the velocity is given by 0.1 times 10 to the 14 times 40,000 kilometers per hour. And this is divided by the mass of the comet. 0.1 times 10 to the 14 plus the mass of the earth, which is 5.97 times 10 to 24. And these results in the variation in the velocity there is approximately 6.7 times stand to minus eight kilometers per hour. Again, a very, very, very small change in the velocity. So the changing the velocity would be very notable. On the other hand, the environmental impact off this collision will be catastrophic and, well, it will be very, very notable. But this question is talking only about the change in velocity, and this is very notable.

Brazilian Center for Research in Physics