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11,3 Sport: shooting Eq: (9.6) gun shoots out a bullet of mass 0.6 02. The gun mass (less the bullet) is 14 02. It recoils with speed of 35 mph. Find the bullet spe...

Question

11,3 Sport: shooting Eq: (9.6) gun shoots out a bullet of mass 0.6 02. The gun mass (less the bullet) is 14 02. It recoils with speed of 35 mph. Find the bullet speed in mph_114 Star Wars travel 2 "dump their garbage before they g0 to light-speed" (Han) A Star Wars spaceship with fuel is initially at rest at point CG Fuel is ejected at the back to propel - the spaceship to the forward direction: After a long time, the total fuel ejected out has center of gravity moving backward at 20 m

11,3 Sport: shooting Eq: (9.6) gun shoots out a bullet of mass 0.6 02. The gun mass (less the bullet) is 14 02. It recoils with speed of 35 mph. Find the bullet speed in mph_ 114 Star Wars travel 2 "dump their garbage before they g0 to light-speed" (Han) A Star Wars spaceship with fuel is initially at rest at point CG Fuel is ejected at the back to propel - the spaceship to the forward direction: After a long time, the total fuel ejected out has center of gravity moving backward at 20 m/s: The mass Of the total ejected fuel is 15 times that of the remaining spaceship. Find the forward speed = of the remaining spaceship relative to CG.



Answers

A Vulcan spaceship has a mass of $65000 \mathrm{kg}$ and a Romulan spaceship is twice as massive. Both have engines that generate the same total force of $9.5 \times 10^{6} \mathrm{N}$. (a) If each spaceship fires its engine for the same amount of time, starting from rest, which will have the greater kinetic energy? Which will have the greater momentum? (b) If each spaceship fires its engine for the same distance, which will have the greater kinetic energy? Which will have the greater momentum? (c) Calculate the energy and momentum of each spaceship in parts (a) and (b), ignoring any change in mass due to whatever is expelled by the engines. In part (a), assume that the engines are fired for 100 s. In part (b), assume that the engines are fired for $100 \mathrm{m}$.

So here we know that the orbital radius is our equaling the radius of the earth plus h the height, the altitude rather and this would be 6370 kilometers plus 400 kilometres. And this is equaling Ah, 6.676 point 77 rather times 10 to the sixth meters. And so for part A using Kepler's law, um, in equation 13 34 we can then find the period of the ships. So the period of the ships would be equal to the square root of four pi squared, multiplied by r cubed, divided by GM. And this would be equal to the square root of four pi squared multiplied by 6.77 times 10 to the sixth meters quantity cubed. This would be divided by 6.67 times 10 to the negative 11th. The units here would be meters cubed per second squared per kilogram and then multiplied by here the mass of the Earth night 5.98 times 10 to the 24th kilograms. And we can say that the period of the ships is going to be equal to 7.68 times 10 to the third meters. My apologies. Um, the period of the ships would be 5.54 times 10 to the third seconds. We can say that this is gonna be approximately equal to 92.3 minutes. Now the speed of the ships for part B, we can say this would be so. The two pi times are the orbital radius divided by the period that we had just calculated. This would be to pi times 6.77 times 10 to the sixth meters. And this would be divided by 5.54 times 10 to the third seconds. And this is equaling 7.68 times 10 to the third meters per second. And so we convince find for part C the new kinetic energy K, this would be equal to 1/2 and b squared. This would be equal to 1/2 times M times. 99% of these ships velocity, quantity squared. And so the kinetic energy would be 1/2 times the mass of 2000 kilograms multiplied by point 99 squared, multiplied by 7.68 times 10 to the third meters per second Quantity Square. And we have that the kinetic energy is equaling 5.78 times, 10 to the 10th. Jules, this would be our answer for part c R Answer for Part B and for part D. Immediately after the burst, we can say that the potential energy is the same as it was before the burst. So the potential energy you would be equal to negative the gravitational constant G, time's and time's m divided by our This would be equal to negative 6.67 times 10 to the negative 11th. This would be meters cubed, divided by seconds, rather per second squared per kilogram multiplied by the mass of the earth. 5.98 times 10 to the 24th kilograms and then this would be multiplied by 2000 kilograms. This would all be divided by the orbital radius of 6.77 times 10 to the six meters. So we find that you that gravitation the rather the potential energy here would be negative 1.18 times 10 to the 11th. Jules, this would be our gravitational potential energy immediately after the burst and then for part you. We then know that it's and it's new elliptical orbit. The energy would be equal to the kinetic energy plus the potential energy. And so this is equaling 5.78 times, 10 to the 10th. Jules Aah! This would be a plus. Negative 1.18 times 10 to the 11th Jules. So essentially, it's total energy would be negative. Six point 02 times 10 to the 10th and the units again. Jules. So this would be our answer for part E four part F. Now we can then say for the elliptical orbit, the total energy can be written as e equaling negative g times sometimes and divided by two A where a is the semi major access. So here they would be equal to negative G times the mass of the Earth times. I'm divided by two times the total energy. This would be equal to negative 6.67 times 10 to the negative. 11th multiplied by 5.98 times 10 to the 24th multiplied by 2000 kilograms, divided by two times the energy that we found in part E. That would be negative 6.2 times 10 to the 10th jewels and we find that a this would be equal to 6.63 times 10 to 6 meters. This would be for the semi major axes, of course, of the need Elliptical orbit on the lips. And so we can then safer part G to find the period. Ah, we can use Equation 13 34. But now we're going to replace our with a And so the period t would be equal to the square root of four pi squared, multiplied by a cubed divided by the gravitational constant times the mass of the earth. This is gonna be equal two four pi squared, multiplied by 6.63 times 10 to the sixth meters Quantity cube. This would be divided by 6.67 times 10 to the negative 11th meters cubed per second squared per kilogram multiplied by 5.98 times 10 to the 24th kilograms. And so the period is now equal to 5.37 times 10 to the third seconds. And we can say that this is approximately equal to 89.5 minutes. This would be your answer for part G four part age now. Ah, we can say that the elliptical, the orbital period rather tea for Picard's elliptical orbit is shorter than eager oars. We can say that the change in the period or the the difference in period would be near it. T not divided by T. And so this would be equal to to 5540 seconds minus 5000 370 seconds. This is equaling 170 seconds. And so Picard will arrive back at point P ahead of Igor s so we can say to card will arrive at point P. Aah! Had of Igor Bye. And this would be 170 seconds minus 90 seconds. Soapy card will arrive at point P ahead of Igor by 80 seconds. This would be our final answer for part H. That is the end of the solution. Thank you for watching

In this problem we have given the expanding gasses that leaves the muzzle off a rifle. Also contribute to the required A 30 caliber bullet. Has a mass massive bullet is given to us, which is 0.0 7 to 0 k g and a speed with with related to the model velocity off Bullard with respect to imagine, he's given 601 yeah, meter per second and when fired from a rifle that has a mass off 2.80 k g. The mass off rifle is given 2.80 k g, and it is given there the loosely held rifle equals at a speed off 1.85 m second right, The spirit off recoil is given 1.85 m per second. If I consider the direction off well, it is positive, then it will be negative, right? So now we have to find the momentum off the properly. Propellant gas is in coordinate system a test, so we know that in the system momentum building comes out. That means initially momentum is zero. That means finally, the total momentum will be zero, the moment of due to bullet do the rifle. And the due to guess is the summation. Vector summation off. These three should be equal to zero. That means PR less BG less BB rifle gas and bullet. The submission off. Momentum off. These three should be called to zero. Now we have to find the velocity off bullet. It will be equal toe 601 minus 1.85 because 601 is with respect to models. So this will be equal toe. Yeah, If we saw this, this will be called Oh 5 99.15 5 99.15 Yeah. 5 99.15 m was second rate. Now we will ride the momentum due to these three. So do toe bullet. The momentum will be equal toe 0.0 0.720 into velocity abilities 5 99 0.15 So this will be equal to around 4.314 k g meter for a second. Right now. Momentum due to rifle, this will be equal toe The result. This will be equal toe 2.80 into minus 1.852 point 80 in two minus 1.85 So this will be equal toe minus 5.18 K g meter per second. No momentum due to guess we can find because the submission off these three are equal. We'll put the value off this and this here and we get the value off mo mentum due to cases. So this will be equal toe if peace all this will be equal toe 5.18 minus 4.3 14 After solving this, we get the moment. Um, do two guesses is equal toe 0.86 kg meter per second, 0.866 Katie meter for a second. And this is positive. That means the direction off this moment, um, will be the same as the bullet. So the direction we can say that the momentum is 0.866 kg meter per second in the direction off in the direction off will it night

We were told a forty five color. Good. If it's shot straight up in the air from the surface of the moon will reach hide a physical. Eight hundred thirty two team minus two points. Sixty square after t seconds. Man, if this was on Earth instead, wait. It's height will be given by s is equal to eight hundred thirty two team minus sixteen. T squared after tea settles, but we want to do It's fine how long the will be aloft in each case and how high this bullet will go. So let's maybe just try a little a picture of what this looked like. The book of these displacement functions are quadratic ce So both of them will look something along the lines of So they both have zeros at zero and then also some past. So they're both look, something kind of like this here and the highest point here. Well, this is going to be where the derivative of s So this is US teeth. And this highest value that we reach her will be where s prime of teeth or are the velocity is equal to who's there? So this is how we're going to find how high the bullet will go. And also we can find how long it will take or how long the boat will be aloft by finding the intercepts of that's the one will be a zero and then this other one here to be our other intercept. So this is a law tight. So first, let's go ahead into this for the moon. So So for the moon, we have that our displacement function is it eight hundred thirty two a hundred thirty two tea. I have to point sixty squared and so first to find a loft, we want to set this equal to zero. So for a loft we have us is equal zero, which means eight hundred thirty two team finest to sixteen squared is a zero. So we can first factor out of tea from this t factored out. And I'm also gonna park Got a two point six, though it's want so two points six and Mr Lee to point sixty that on factor goat two points sixteen and then a hundred thirty two divided by two point six is three twenty and then factor out two point sixty. From here we'LL just get team. So this tells us by the zero product cocky that tea is equal to zero or tea is equal to three hundred and twenty. And so both of these are supposed to be in seconds for units, since that's what air time with him. So the total time to be a loft on the moon would be three hundred twenty second, since tea is equal to zero is when we first shoot the bullet and so we can get rid of take it easy. And now, to find how high the bullet hates to go. So, like we're saying, Well, here, we want to set s pride, Ego Zero or our Velocity Zero and then solve for that value released for the time, and then we can plug him. So now we want to find Max height, and we're still on the moon. So we need to find what the is going to be. People squiggly line to lease. So velocity is going to be the derivative. Oh, this expression here. So we confer First, break this house eight hundred thirty two DDT, but tea violet color minus two point six. Do TT of T square give one to black sort of green Hey, now the drilling of tea will This is T to the first power So we would move the one out front using power went off because from blue we'LL get out front contract so that would be eight hundred thirty times one which would just be a hundred and thirty two and then teach zero power will be one So we don't have to right there and then like wives over here, we can multiply. Cool. You can multiply that two front using our rules truck. Come on. I want that with negative by point two. And this will be tea of first power. So this this So now we want to buy what makes this equal to zero since we're looking for this horizontal tangent line of our displacement function and so we can go ahead and add over keep us too. So we get put implies So a hundred thirty two five two teeth and then dividing each side by my point too. We get that tea is one hundred and sixty seconds, so I'm sure we don't need the time. Well, we want to know is how high it's going to go. So we're now going to take this value and plug it into s so we can actually find what that output it's. So now we want to look at what s of one hundred sixty years on, so this will be so We get a hundred two times one sixteen and there was tracked on two point six times, one hundred and sixty squared. And this is going to be sixty six thousand five hundred and sixty. And you know, there's going to be for this. Here is how high is it going when we shoot it? And now this answers everything under the moon conditions. So we're going to repeat the same process for stuffer. And so I'm going to do this on the next page because I don't have a room. It'LL be a sad expression, just minus sixteen. So Earth eight hundred and first, this is between eight hundred thirty two, minus sixteen. Team square. No TV here. No, it's just unsure of the damn car. They make a great team to sixty square. Good. Now, for the first part, we want to hide a loft. And so remember, this is supposed to be the total time it's in the air. And since this here look something similar to this words craft, wherever it crosses the X axis for a second time will be our total time A lot. And so let's go ahead and set s equal zero. So we want to do s equals zero, which is going to be eight hundred and thirty two t minus sixteen. Squared is a zero. And just like we did before, I'm going to pull a lot of Tia's welds, coefficient or t squared. So we're gonna look at sixteen t Hi, I'm so let's see, eight hundred two divided by sixteen is fifty two. So fifty two minus T secret zero and once again by the zero product property. This implies that tea is zero or tea is equal to fifty two and we can get rid of teasing with zero for a similar reason for and I'LL just go ahead like our units there for seconds. So this bullet here on the earth will be a lot for fifty two seconds now to find our other piece of this. So lastly, to find our max long height So next Sonam pipes or this, and recall that the marks are high is going to be where we have a horizontal tangent line for this which is where s prime of tea which is a good whore Blossom is he? So we want to take the derivative of us so we might possess us is equal to eight hundred or two times the derivative with respect t of teeth minus sixteen. Derivative expected he but t squared. And just like before, both of these derivatives will be pi Powerball. So this has the power of one who will move this help front struck one off of it. So eight hundred thirty two multiplied by one was just going to be a hundred thirty two And then Tito's here. How will also be one? So we don't have anything to write And here we would move to outfront power will get trucked off one and wonder with negative thirty two and we'll still have t But I will be to be first power and we want to find where this is equal to zero because we're trying to find this horizontal line here. And to do that, we would add thirty two over. So this six to eight hundred thirty two to thirty two teeth and then we divide each side by thirty two, which is going to be tea. Is it too? Twenty six my unit's for This will be in seconds, and I want to do the exact same thing as last time. So we found the time. And to actually find the height, we want to take this value and plug it into us now. So us of twenty six is going to equal too. So let's see. So I have eight hundred thirty two times twenty six and minus sixteen times twenty six squared, which is ten thousand eight hundred and sixteen don't. And so this is the maximum height for our bullet on Earth. And so the answer is, we get here should look about right, since we know that the gravity on the moon is less than on Earth, so it makes them stop the bullet. Honda Moon is going to be in the air for longer as well as reach a higher height as opposed to the Earth

So they give us thes position functions for the height of this bullet on Earth and the moon, and they first want to figure out how long each bullet will be a flute And how high will the bullet go? So let's go ahead and figure out how long will be afloat, because that's essentially like saying so when we shoot this bold up in the air, the it's gonna sell maximum of this going to kind of come back down like that. And so in both of these cases here, when we're on the ground s is really just equal to zero. So we just come over here. I shall do this in red, uh, set this ego zero then self, so we can factor on T from this Sophie tee times 8 32 minus 2.6 t. And so by the zero product property, either tease you could zero or 830 to minus 2.60 is equal to zero. And we could throw out, uh, t being zero. Because that would mean like no time has passed. So we haven't thrown the ball yet. I mean, we haven't shot the gun yet. so over here we go and solve. So we'd add that over. So 8. 32 music to 2.16 and then divide 2.6 over. So 8. 32 divided by 2.6 is 320 t and whoever the units for this seconds. Okay, so this would be 320 seconds. Um, for how long? It will stay afloat for and then over here to figure out on Earth, we do the exact same thing. So we just said to see equal to zero factor off the tee so it would be tee times a 30 to minus 16. T is equal to zero. And so again, we can't have Ortiz. You could zero, so we don't need to write that one down. We could just do 8. 32 then we died. The 16 over. It would be 16. 18. Now we could divide each side by 16. So 8, 32 divided by 16 is 52. So this is going to say T is equal to 52 seconds. So this is how long it's going to take for the bullet to go up and then land again on Earth. All right, Now, the next thing they say is wow, find the maximum height the bullet reaches in each case. So there's two ways we could do this since our position was given by a quadratic, we could just find the vertex for it. But remember, calculus, um, well, tell us where this function up here, Like this kind of sketch of what we have is that its maximum? Because remember, there's essentially a rate of change of zero there, or I guess, in the more physical way to think about is our velocity become zero at that point. So if we take the derivative of this city grew zero that will tell us what our maximum height occurs. Then we just plug that back in. So let's go ahead and do that. So we need to take the derivative of us because V is equal to d s by D. T. So you d by d. T of 8. 32 t minus 2.60 squared. And so remember we can go ahead and distribute this across addition and subtraction, and we could also pull out any Constance so we can rewrite this as 8 32 D by D T of T minus 2.6 D by d. T. Of t. Squared Now the derivative of tea is just going to be one. And then to take the derivative t squared, we use the power rules that remember, we pull the two outfront subtract one off, so that's just gonna be to t. So now we can multiply all that together. Those will be 8 32 minus 5.2 t Remember, we want to set this equal to zero now, so I'll do that in a black. So said that he would zero and then we just offer tea so it would be 5.2. T is equal to 8. 32 and then we divide 5.2 over. So 8 32 divided by 5.2 is 1 60. And that kind of makes sense because the time it takes for us to get from, um, the ground back to the ground again is 3. 60. I mean, 3. 20. So the time it would take to reach the maximum, we might suspect to be half of that. Um, right. But now we just need to go and plug this into here and they'll tell us the Mac side. So we will do s of 1 60. So that's going to be and I actually wanna factor that t out. So it would be 1 60 times 8. 30 to minus 2.6 times 1 60. And then if we plug this and so negative 2.6 times 1 60 at that 8. 32 then multiplied by 1. 60. So it would be 66,560. And what units were these in again? This was in feet. So this is going to be the max height, Okay? I went a little crazy, bitch. So that's our max height, at least on the moon. Now we come over here and I do the same thing. So let's see, We get V is equal to d s by D. T. Yeah, and so then we take the derivative of this, and actually, I was gonna go ahead and distribute it right away so it would be 8. 32 D by d t of tea minus 16 D by d t of t squared that we already said, Well, both these derivatives, or that's just one. This is to t multiply everything together. So we get 8. 30 to minus, um 32 t, and we want to say this equal to zero. And so then that would give us 32. T is equal to 8 30 to divide each side by 32. So 8 32 divided by 32 is 26 seconds. And again, the observation we made about it being half of the time for how long it was in the air for all together makes sense. Um, but now let's go ahead and put this in to our position to 26. And so it'll be 8 32 or I shall pull out that t once again. So it'll be 26 times 8 32 minus 16 times 26. So negative 16 times 26 at that to 8. 32 then multiply that by 26. So this is going to be 10,816 and this should also be in feet. And so this is going to be the max height on earth. And so that kind of makes sense that it's like six times the difference. Um, because I think the acceleration on the moon is supposed to be. I think it's like six times as small and like the difference of the heights, or about six times as well. So that kind of matches up as well. Eso not really needed that last thing, but just kind of observation about the Max Heights for those.


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