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Problem 112 Easy Difficulty

For the rocket described in Examples 8.15 and 8.16 (Section 8.6$)$ , the mass of the rocket as a function of time is
$$m(t)=\left\{\begin{array}{ll}{m_{0}} & {\text { for } t<0} \\ {m_{0}\left(1-\frac{t}{120 \mathrm{s}}\right)} & {\text { for } 0 \leq t \leq 90 \mathrm{s}} \\ {m_{\mathrm{d}} / 4} & {\text { for } t \geq 90 \mathrm{s}}\end{array}\right.$$
(a) Calculate and graph the velocity of the rocket as a function of time from $t=0$ to $t=100 \mathrm{s}$ . (b) Calculate and graph the acceleration of the rocket as a function of time from $t=0$ to $t=100 \mathrm{s}$ (c) A $75-k g$ astronaut lies on a reclined chair during the firing of the rocket. What is the maximum net force exerted by the chair on the astronaut during the firing? How does your answer compare with her weight on earth?


a. $3327 \mathrm{m} / \mathrm{s}$
b. $a=0$
c. $6.0 \times 10^{3} \mathrm{N}$


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Video Transcript

{'transcript': "So in this case you have a rocket that's going up. And it is also ejecting some mice here at a certain rate. And you want to Dr Equations for velocity acceleration. And then finally the position, Compare them to what you would give it. So you're given that the velocity of the exhaust, its on 1500 commuters perspective and the changing master times with the M T here is 2.5 and then you're told that the time that it takes to lose or the mass here, which is t these are on their shoes. Seconds. The first thing you want to do is find the expression with velocity. So we that expression is already has really been driving. That's equation 943 in the tractor. And that impression says the final minus the initial is you go to the velocity of the sauce times the nicer log. I m I over so already you can see that we have an expression that looks similar to what was given in a problem. Oh, so now well, you have to do now is, um, substitute some of the values that we had that will give it so that we can make this look like the answer. So they tell us that this is equal to the rate K over here home, and they tell us that this staying starts from rest. So the I is zero. So that means that this term goes to zero with me is the final velocity is actually equal to E g national log so noticed the initial mass is given and the initial mass is actually emits when you said that we want So we actually want to remove the final Masum here. But we know that it's changing at this rate. Then the massive any point in time is going to be the initial mass minus the total mass that we have lost year, which is the rate of change of mass times time. So this is just going to be OK. She and that gives you an expression for the Boston. But to put it in the form that's given, you have to reverse this log here. So that's just a matter off putting a negative sign here. So that becomes negative. B nicer log. And I want this key to over. So now it's just do so if you flip this, then you have to have a negative sign on, because it's look, Am I minus? The denominators changed the the positions of these Europe to take on a negative sign outside. So notice that this can you simplify further. So that's e natural log. Oh, this is one my eyes. Okay, t over and I and that's your velocity there. And then they ask you to plot what that velocity would look like. So, essentially, now, when you want to do is put in the numbers that you were given in this case, I think the last thing that they give you said the initial mass was around 13 60 kilograms. So in this case, you look at this equation to substitute or the bodies that you would give him. So you know. Okay. You know, am I and you know he So you're going to have an expression that looks something like 1500 natural log one minus 2.5 over 3 60 times t. So we cannot wait. No, you So you can use your favor. Applauding program will just get a bunch of points and plot them. And you see that from zero to around 1 32 seconds Is the time window that you've given So the when the time is zero here. Nor he said, this is the law of one which is zero. So this has one point up here and for very lives, values of time. So if this time is really, really large here, notice that this is one minus when it comes. So actually, if you look at this when this is, you go to one. So when two point tease you go to 3 60 a natural law is actually natural. Ahb zero nationalize you're blows up. So these things actually going to go up like this and you can put it in the numbers and find the points you want graphs should look something like this and then So now for part B, you have to come up with an expression from the acceleration. So know the acceleration is always going to be the derivative of velocity with respect to time. So in this case, we take our expression here, which was native the E J log wan t over and hi in this case, and then want to take the derivative well that so that is going to be so. This is the constants which stays is Oh, but before you do so notice If you were a natural law, you take a derivative. This is like a changeable here, so derivative of the inside is just going to be okay over and bye bye. There's a negative sign here with negative sign years with that takes away that neighbors high. So you just have something like that. But now when you do it due to the natural law, this is going to be one over whatever is inside that nitro long. So that's your expression for the acceleration, But that can be simplified for their off into something that looks like a V he over your march by this mass in here you get m high muddiness, Katie, and that's your solution expression which matches what's given in Sufism. And again, you have to thought that from may be easier to wanted to do something year. Oh, so again you do the same tests, Um, when t zero Oh, so when this t zero Northey said this is a some non zero volume here Oh, he gets excision, doesn't he's not going to stop exactly zero because t zero this expressions so that when we start something that's here, but then as t eeg resistance and I may get smaller so it means this more volume goes up. So this can actually look like someone in this, but it's not exactly a lot, but it's just going to So then what? See, this thing is asking you now to find expression for the position off this thing. Um, and if you want to boot to grow position, I get you have to drive inspiration for the position. But we know that the position, yes, final exit initial, whatever it is, your foot here is going to be integral with velocity. With this back to add in the world, we have seen about the ferocity with respect to time. So this will give you, oh, the position or changing position because we know that Oh, the velocity is a derivative. Off positions to go backwards if you integrate the expression. So we already have an expression for the Boston, so want to integrate that expression. So this integral from zero t o negative e natural log one. But this key T over and I in this case, so you can actually look up in a row in the table. Oh, and one that you want to use. Here's something that looks like this natural law of pigs be here. This integrates into Oh, x plus Be over a So you're gonna look out this injury? There's no need to redo it. It's already given somewhere. That's a eggs. Let's be but Miss X here. So this expression knows exactly like the expression you have. So we just substitute a with negative Kate, you m and B one. And then you see that the final expression they get Well, look something like So this would be Marx buying everything inside bracket another. Am I? Okay. This tea, this is marked by the natural law. Warren Marlys Katie over in by plus t. So notice year B is actually k over em. But this is oh, so he wants you once we evaluated No, no, a is k over so one of A's family again. So that's why this is Oh, this is written this way because the negative sign that you had before changes the sign here and it's inside here into this is your final expression, porosity for the position which matches what you are given a swell. And you can notice that when t zero this is zero. This is your So when t zero every year you don't have there was not concentrated. Oh, so then now you just want to block this thing, and I'm just going sketches. Zero wanted to again. Oh, I notice again that so when t zero these things that zero like you said before, um and then this thing just increases, but it so again, an increase in the same manner. But this dominates his natural law. So you see something that looks like a national are going up, and that will be your So this review was this invests time grab. Oh, with that problem."}

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