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If a rock is thrown vertically upward from the surface of Marswith velocity of 20 m/s, its height (in meters)after t seconds is h = 20t − 1.86t2.(a) What is t...

Question

If a rock is thrown vertically upward from the surface of Marswith velocity of 20 m/s, its height (in meters)after t seconds is h = 20t − 1.86t2.(a) What is the velocity (in m/s) of the rockafter 4 s?_______- m/s(b) What is the velocity (in m/s) of the rock when its heightis 45 m on its way up? On its way down? (Round youranswers to two decimal places.)up _______ m/sdown ______ m/s

If a rock is thrown vertically upward from the surface of Mars with velocity of 20 m/s, its height (in meters) after t seconds is h = 20t − 1.86t2. (a) What is the velocity (in m/s) of the rock after 4 s? _______- m/s (b) What is the velocity (in m/s) of the rock when its height is 45 m on its way up? On its way down? (Round your answers to two decimal places.) up _______ m/s down ______ m/s



Answers

If a rock is thrown vertically upward from the surface of Mars with velocity 15 m/s, its height after $ t $ seconds is $ h = 15t - 1.86t^2. $

(a) What is the velocity of the rock after 2 s?
(b) What is the velocity of the rock when its height is 25 m on its way up? On its way down?

This problem Number fourteen of this tour Calculus Eighth edition, section two point seven If Iraq has thrown upward on the planet Mars with the velocity of ten meters per second, it's height in meters after tea. Seconds is given by H is equal to ten t minus one point eight six t squared part a Find the velocity of the rocket After one second party find the velocity the market twenty equals a party When will the rock hit the surface and party? With what velocity will the rock hit the surface? So since we're asked to find the velocity at a few different times I including in part B where we're looking for the velocity in a general As a general expression, we'LL begin with part B and used that to solve part eh, auntie on Then proceed two parts as well. So for part B on the Ross iti of this ah rock is going to be the slope function for this position function that was given the height function h so it'll be in the limit s o the velocity Well be that limit as h of purchase zero are the function evaluated at a plus age tee times the quantity of his age when it's one point eight six. Time's a quantity. April's H squared minus this equation evaluated, eh? Tea Time's a minus one point eight six A squared. Hey! And then all this is over each. Our next step is to simplify the numerator we have here today. Liston Age care when we factor out April's H or when we distribute foil Ables Age squared. We had a squared plus two eight, eight plus h squared. So wait, we're going to have Linus twenty points a squared and we're in. The next turn is to a so two turns. No. One, three point. You some are. Quick You, eh? Minus one point each. Squared minus ten, eh? Class one point eight six Peace. Where? All over h here we see positive. Ten million. The numerator and negative penny goes to counsel. Negative one point sixty eight squared. Haunted at one point eight six feet. It's where Cancel the remaining terms or ten age minus three point seven to eighty H minus one point eight six each. Sward, we're gonna cancel lead teaching the denominator with one H each turn and then to evaluate the limit. We we will see that the terms that have a nature meaning such as this Terminator one point each vanishes H depictions of zero. So, in fact, when we get it left over ten minus three point seven two, eh? Those are the only terms that are, uh, they do not have an agent Did not finish and not counselor in this previously. And so this is our answer part B. The velocity of the rock twenty equals a Let's return to party and use this result for this. What is the policy of the rock after one second? Well, once I can t equals a people won, the time will be or the velocity will be ten minus three point seven, two times one You, too. Six point two eight meters per second. That is the answer to part one eight Murphy. We're we're using this equation for the Ross iti at any time equals equals fourteen. When will the rockets surface? This has to do with the hind equation on Iraq is at the surface when h equals zero. So when there is no right to the very core temp T minus one point six, you squared. We can, uh, factor at eighteen. Commander attempted central offering. And so fourteen. Yeah, t times the quantity t minus one point eight sixteen. And we see that the answer for this equation aren't equals zero. They're also twenty calls our ten year edible and the writer by one point eight six. Approximate for every point three seventy. So what this tells us is that the rock is on the surface at two times at T equals zero and a tea cools ten divided by one point six, which is five point three seven six. This makes sense because he is here with the initial time when it was thrown and then five point three seven six seconds later, it is back on the surface. Finally, party. What will the velocity B for the rock when it hits the service? Well or any of your equation. Ten minutes, three point seven two pay. And we need this time five point three seven six when we know it hits the surface and we're gonna find a velocity this way. B minus V equals ten minus three points too. The time approximately five point three seven six. And if we saw this. We actually get almost exactly native ten. We should get exactly negative Tim in effect. And the reason for this is there. The initial velocity of the rock at T equals zero was ten meters per second upward. Positive velocity twenty seven seconds later. It is back at the same height and is travelling in the opposite direction so it will haven't seen Speed ten seems to be ten, but in moving in opposite direction, which which is consistent with the native allies. So the answer her party is negative. Ten liters per cent.

So rock is tone upward on Mars. The velocity, you know of turn user's personal and its height. After two seconds, he's given by Interesting the holes. 10 to blindness 1.86 She squared no a You want the velocity of the rock after tea scents The velocity is given by derivative, but I Well, that's turn my nose. I'm gonna take There's two You know it's a cracked one from it That's few terms. 1.6 to first. Yes, Christians. 10 my ass. 3.7 two. Hey, the velocity at any point in time, eh? Well, he's gonna plug in a pretty 10 lines. 3.7 two See? Is final the rock at the ground. That's when agent you zero So a treaties remember 10 T minus point point six and he squared nasty equal zero negative t zero. Then the equation sense. But we know that the rock starts from the ground. Then it's thrown up. So we need a point further in time, so T is not equal to zero. Then we can divide by two. We'll get 10 minus 1.6 t zero acquittal in me sand equals one point a 16 or equivalently t is equal 2 10 divided by 1.6 no. D That's to find the velocity at that point. Team have just found his calling. If you won velocity t one, it is turning miners 3.7 two 21 which is turn miners two times, t one. Because we're just dividing 3.7 to 1.86 That gives us his too two times 10. This is a course negative turn meters.

So before we actually do a, um I would make note that it would probably be better to do be first and then just use the result from B to plug in a, um, but the way they have it set up, they want us to do this first and then to go from there. So I'm just going to go ahead and follow the steps. But if you want, you can just kind of skip ahead, see what we do and be, and then just plug one into that value, and that should give you the same thing to get over here. I mean, I'll save you a lot of time, but just kind of for the sake of how they have the problems that I'm going to kind of go from a then b so over here, remember that the velocity, if we think about it, is just the derivative of our position. So in this case, H prime of T is going to be good a v of T. And we know and maybe I should write out what the definition of this is first. So H prime of T is going to be equal to the limit as H approaches zero of h of T plus H minus H of t all over H, yeah. And so in this case, if we want one, then we're going to replace those teas with one. Uh, so, um, since this is the same thing as beauty, we can just go in and write that. So it would be V of one is two the limit as H approaches zero of capital h of one plus little H minus h of one all over little age, And now we can plug in one plus h into here as well as one. So that would be the limit as a church approaches zero so h of one plus h. So that would be 10 times one plus each and then minus 1.861 plus h squared minus. Well, if we come over here and plug in one to be 10 minus 1.86 and then this is all over each, actually, I'll need to scooch this down a bit more. It looks like, um, then we can go ahead and distribute everything. So this would just be 10 plus 10 h. Uh, this here is going to be wolf. We expand that out, it would first be one plus two h plus h squared. And then we would distribute the negative 1.86 So I'll go ahead and do that. Um, And actually, before we do that notice first this 10 and that can cancel out with each other. And then if we distribute this negative 86 this one and this, we'll cancel because we have negative. They're absolutely those Cancel out. So, actually, let's write that out just so you can see what we have left before I actually started doing stuff to be 10 h minus 1.86 times two h plus h squared all over h. And then we can go ahead and simplify this down by combining our like terms. Um, so we'd have 10 minus, actually. Let me distribute that 1.86 for 1.86 times two. So this is going to be minus 3.72 and then minus one point 86 h squared. There should be an h there that we can combine those so 10 minus. That would be 6.28 So we have the limit as h approaches zero oh, 6.28 H minus 1.86 h squared all over h. And now notice we can divide this agent to each of those terms to just give the limit as H approaches 6.28 minus 1.86 h. And then lastly, we can go ahead and apply our limit. Um, that's just going to turn this term to zero, and we're going to be left with 6.2. And actually, what are the units with this, uh, meters per second. So this would be meters per second. So our velocity after one second is going to be 6.28 m per second? Yes. Now we can come over here and repeat all these same steps, but in general. So, um, instead of replacing t with one, we're just gonna replace it with a So let me just go ahead and race this here. So this should be the limit. As H approaches zero of capital h of a plus little H minus capital h of little A all over little H. And then again, we can go ahead and plug a plus h into here. So this is going to be actually music at this overall. But I think I'll need it a bunch of space. So the limit has h approaches. Zero Oh, so would be 10 a plus h minus 1.868 plus h squared and then minus 10 a minus 1.868 squared and then this would be all over each. So again, let's go ahead and distribute. So distribute the 10 to be 10 a plus 10 h here. If we expand that out, they'll give us a password. Plus two a h, I don't know, plus h squared. And now one thing to notice again is this. 10 a here councils out with this 10 a And then once we distribute this negative 1.86 this a squared is going to cancel with that a squared. And if we write this all out, we would end up with 10 H minus 1.86 times two a h plus h squared all over h and then we can go ahead and just distribute that negative 1.86 It would be the limit has h approaches zero of 10 H minus 3.72 a h and then minus 1.868 squared all over h. And now, once again, we noticed we can divide H into everything in the numerator. And I guess one thing I should say is when you're doing the derivative, if for some reason you just can't divide out the HR, at least for most of the ones that you'll probably be interested in doing, at least for this first kind of time of you learning this, uh, you mess something up. So if you can't just, like, divide this age evenly, then you may need to go back and look at your steps again. Come because otherwise you just can't really do anything. Um, What's that? Yeah, uh, now we can't combine anymore like terms, So let's just apply the limit. Well, now that is just going to become zero. So we'd end up with 10 minus 3.72 a. And this is our velocity in general. And so again, if you were to come over here and just plug one into this like I was saying, you would get the same value. Um so yeah, again. Doing this would have saved you a bit of time, but just for kind of sake of how they have the problem set up and I guess practice. I went ahead and did a first anyways, against my better judgment. Right now for the next part, let me scoop this down. We want to figure out when the rock hits the surface. Well, that's essentially just saying, When is our height zero? So this is really saying H O. T. Is equal to zero when so we can just come over here. That's the secret to zero will factor that So b. T times 10 minus 1.86 t is equal to zero. So that tells us t is zero or 10 minus 1.8 66 0. And so t is equal to zero. Doesn't really mean anything for us in the sense of hitting the ground, because that's when we're throwing it initially so we can throw that one out, and then we just solve for this. Um so we would add the 1.86 over and then divide to be T 0 to 10/1 00.86 which would give us so first, Exactly the answer is going to be 500 over 93 but I have no idea what that is. So that is approximately 5.3 seven. And actually, that should get rounded up because the next number is a six. So about 5.38 seconds. So this would be the time it would take in order for us to hit the ground. And then if they want us to find what is the velocity when the rockets the ground, Then we can just take our formula that we got up here. We screw this down and then plug in our time, so we're going to plug in 593rd into this. So this is going to be our velocity at 593rd is going to be 10 minus three point 72 times 593rd. And then if I go ahead and plug this into my calculator, uh, 10 months that well, times and minus 20. Uh, that would give us negative 10, and then it would be meters per second. So this would have been our speed once we hit the ground. Which makes sense because we're moving downwards now.

So before we actually do a, um I would make note that it would probably be better to do be first and then just use the result from B to plug in a, um, but the way they have it set up, they want us to do this first and then to go from there. So I'm just going to go ahead and follow the steps. But if you want, you can just kind of skip ahead, see what we do and be, and then just plug one into that value, and that should give you the same thing to get over here. I mean, I'll save you a lot of time, but just kind of for the sake of how they have the problems that I'm going to kind of go from a then b so over here, remember that the velocity, if we think about it, is just the derivative of our position. So in this case, H prime of T is going to be good a v of T. And we know and maybe I should write out what the definition of this is first. So H prime of T is going to be equal to the limit as H approaches zero of h of T plus H minus H of t all over H, yeah. And so in this case, if we want one, then we're going to replace those teas with one. Uh, so, um, since this is the same thing as beauty, we can just go in and write that. So it would be V of one is two the limit as H approaches zero of capital h of one plus little H minus h of one all over little age, And now we can plug in one plus h into here as well as one. So that would be the limit as a church approaches zero so h of one plus h. So that would be 10 times one plus each and then minus 1.861 plus h squared minus. Well, if we come over here and plug in one to be 10 minus 1.86 and then this is all over each, actually, I'll need to scooch this down a bit more. It looks like, um, then we can go ahead and distribute everything. So this would just be 10 plus 10 h. Uh, this here is going to be wolf. We expand that out, it would first be one plus two h plus h squared. And then we would distribute the negative 1.86 So I'll go ahead and do that. Um, And actually, before we do that notice first this 10 and that can cancel out with each other. And then if we distribute this negative 86 this one and this, we'll cancel because we have negative. They're absolutely those Cancel out. So, actually, let's write that out just so you can see what we have left before I actually started doing stuff to be 10 h minus 1.86 times two h plus h squared all over h. And then we can go ahead and simplify this down by combining our like terms. Um, so we'd have 10 minus, actually. Let me distribute that 1.86 for 1.86 times two. So this is going to be minus 3.72 and then minus one point 86 h squared. There should be an h there that we can combine those so 10 minus. That would be 6.28 So we have the limit as h approaches zero oh, 6.28 H minus 1.86 h squared all over h. And now notice we can divide this agent to each of those terms to just give the limit as H approaches 6.28 minus 1.86 h. And then lastly, we can go ahead and apply our limit. Um, that's just going to turn this term to zero, and we're going to be left with 6.2. And actually, what are the units with this, uh, meters per second. So this would be meters per second. So our velocity after one second is going to be 6.28 m per second? Yes. Now we can come over here and repeat all these same steps, but in general. So, um, instead of replacing t with one, we're just gonna replace it with a So let me just go ahead and race this here. So this should be the limit. As H approaches zero of capital h of a plus little H minus capital h of little A all over little H. And then again, we can go ahead and plug a plus h into here. So this is going to be actually music at this overall. But I think I'll need it a bunch of space. So the limit has h approaches. Zero Oh, so would be 10 a plus h minus 1.868 plus h squared and then minus 10 a minus 1.868 squared and then this would be all over each. So again, let's go ahead and distribute. So distribute the 10 to be 10 a plus 10 h here. If we expand that out, they'll give us a password. Plus two a h, I don't know, plus h squared. And now one thing to notice again is this. 10 a here councils out with this 10 a And then once we distribute this negative 1.86 this a squared is going to cancel with that a squared. And if we write this all out, we would end up with 10 H minus 1.86 times two a h plus h squared all over h and then we can go ahead and just distribute that negative 1.86 It would be the limit has h approaches zero of 10 H minus 3.72 a h and then minus 1.868 squared all over h. And now, once again, we noticed we can divide H into everything in the numerator. And I guess one thing I should say is when you're doing the derivative, if for some reason you just can't divide out the HR, at least for most of the ones that you'll probably be interested in doing, at least for this first kind of time of you learning this, uh, you mess something up. So if you can't just, like, divide this age evenly, then you may need to go back and look at your steps again. Come because otherwise you just can't really do anything. What's that? Yeah, uh, now we can't combine anymore like terms. So let's just apply the limit. Well, now that is just going to become zero. So we'd end up with 10 minus 3.72 a. And this is our velocity in general. And so again, if you were to come over here and just plug one into this like I was saying, you would get the same value. Um, so Yeah, again, Doing this would have saved you a bit of time, but just for kind of sake of how they have the problem set up. And I guess practice. I went ahead and did a first Anyways, against my better judgment. Right now for the next part, let me scoop this down. We want to figure out when the rock hits the surface. Well, that's essentially just saying, When is our height zero? So this is really saying H O. T. Is equal to zero when? So we can just come over here So the secret to zero will factor that So b t times 10 minus 1.86 t is equal to zero. So that tells us t is zero or 10 minus 1.8 66 0. And so t is equal to zero. Doesn't really mean anything for us in the sense of hitting the ground, because that's when we're throwing it initially so we can throw that one out, and then we just solve for this. Um so we would add the 1.86 over and then divide to be T 0 to 10/1 00.86 which would give us. So first, exactly the answer is going to be 500 over 93 but I have no idea what that is. So that is approximately 5.3 seven. And actually, that should get rounded up because the next number is a six. So about 5.38 seconds. So this would be the time it would take in order for us to hit the ground. And then if they want us to find what is the velocity when the rockets the ground, Then we can just take our formula that we got up here. We screw this down and then plug in our time, so we're going to plug in 593rd into this. So this is going to be our velocity at 593rd is going to be 10 minus three point 72 times 593rd. And then if I go ahead and plug this into my calculator, uh, 10 months that, well, times and minus 20. Uh, that would give us negative 10, and then it would be meters per second. So this would have been our speed once we hit the ground. Which makes sense because we're moving downwards now.


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