Problem

$\bullet$ $\bullet$ When a piece of wood is press…

03:12

Problem 50 Hard Difficulty

$\bullet$ $\bullet$ A slingshot obeying Hooke's law is used to launch pebbles
vertically into the air. You observe that if you pull a pebble back
20.0 $\mathrm{cm}$ against the elastic band, the pebble goes 6.0 $\mathrm{m}$ high.
(a) Assuming that air drag is negligible, how high will the peb-
ble go if you pull it back 40.0 $\mathrm{cm}$ instead? (b) How far must
you pull it back so it will reach 12.0 $\mathrm{m}$ (c) If you pull a pebble
that is twice as heavy back $20.0 \mathrm{cm},$ how high will it go?

Answer

a) 24 m
b) 28.3 cm
c) 3.0 m

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Physics 101 Mechanics

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Sharieleen A.

October 26, 2020

Finally, the answer I needed, thanks Abhishek J

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Video Transcript

Okay, so we have a slingshot which is all being hooks law. So let's see that the sling so sleek shot is like a spring. And we have a people that we put it on top. And then what we do here is we compress the spring so that it bounces back on because the people up in there, so it will go for particular up. Because if we are compressing and vertically downwards, so technically it should go straight up. Now we're given that if we compressed the spring toe 20 centimeter, then the people three teas and height, it is a hider six meters. Great. So that's thie uh, compression. And that's the height. Now let's try to draw that in areas. Well, So when the spring is compressor, let's say that's when we compress the spring. The people critters around here on it is said that this high 86 meters now we can immediately see one thing that we can use. Ah, the energy conservation to solve this problem because we know that there will be some potential energy that is stored due to the compression. Andi. Also, when the people goes up, then there will be the gravitational potential energy that were stolen, the people and also regarding the kinetic energy. Initially, the Canadian is you'll be zero around here because the initial velocity zero. But then sorry. There'll be some initial velocity because that's what Ah ah, throwing the people upwards. But at this point, the final velocity will be zero here. So let me write that down in the mathematical sense. So what we're saying is we have the final candidate unity. So let's call this point f and let's call this point our let's go When the people was here, let's call that point as I or the initial point so potential energy and to find any at the final point, plus the plus the Kennedy Energy. The final point is equal to the Kennedy Energy at the initial point, plus the potential energy at that point. Now, as we said that and defining point here, our potential energy is essentially the gravitational potential energy with his m times, g times H. So we can easily write you f as MGH. Also, since the velocity zero there, we can safely ignore this part. Right. Um and one more thing is, since we're starting with a compressed situation. And when we're trying to throw it, we can say that initially the velocity is zero here just the moment before we start trying it when we're compressing it, we don't have any velocity, right? So that's why we can say that this part is here as well. But of course, after we release a spring, their reason velocity. But we're not worried about that because we're only dealing with the initial and final point. And as we mentioned that initial point people is addressed. So that means there's no put a Canadian energy there. But of course there will be some potential energy. Do you tow the spring being compressed? So do you do the compression of the the spring? The potential energy will be half K excess squared where excess square being B exiting the amount of compression that the spring had. So, for example, we had a 20 centimeter of compression. Now from their weekend right instead of u F, we can write MGH, and instead of the Y, we can write half que existe squared. So that's the relation that were found using the energy conservation. Now let's solve the problem part by part sin, part A. We need to find out that if the spring is compressed 40 centimetres So let's actually call x one on X We se x when he's 20 centimeters, which is the given value. And we need to find the amount of hide that the people reaches when we compress it to four centimeters. So we call that x two. So as we derived here this equation right here, we wrote it down here. Now, if we actually manipulate this equation a little bit and see that if we put Xs squared on the left and take mg time on the right, then we have this whole thing constant because we're not changing the weight of the mass of the people. So this mass gravitational, constant and key or the spring concert is constant for both of the scenarios. So this whole thing is constant, so we can easily say that the ratio of eight and excess square is a constant er, so that's constant. Then we can make toe. We can take two scenarios off this ratios and then we can equal both of them. So this is when the spring is compressed by 20 centimeters and this in areas when the spring is compressed by 40 meters. And since this party's Constance or no matter how much we're compressing, the ratio of height over the compression squared will be constant. Always. So that's why we sent these two conditions equal to each other from there. If we solve for the height. So H two is the high that the people will reach after we complicity for his animators. So each toe is the height for 40 centimeters compression, right? So if we saw for aged too, we see that we'll have 81 times extrovert x one squared. Ana, We know all the values over there. So we plugged those in and see that even reached 24 meters off right in part B. We're saying that we need to find out the compression this time because we're given the height of the, um, people. So what we're given here is we're saying that okay, the people will reach 12 meters Then how much compression do we need to read toe pull? How much compression we need toe put the people at 12 meters. Now we'll be again using this equation right here. but then went in some things lake instead ofthe. Since this part is constant, but now we need X two or the compression so we can set extra in terms of the other toa terms. So that means write that down. So from here we know that h one over X one squared is equal to H two over x two squared and we're solving for X to. So if we do so, we see that X two squared is equal to x one squared times. It's too over h one and from there, if we take square root on both sides will have extra which is equal to X one times square. Root off is to die each one and that's what we right here. So we know that the final height will be 12 meters and we're already given the condition which is the falling were like if you compress it spring 20 centimeters it read six meters so we can use that piece of information in excellent and h one perimeters. Now from there it's all for it. We see that the compassion should be 80 28 countries animators So if we pull back 28 20 centimeters. The people will reach meters Now. Finally, we need toe the now the fight. Finally, what is changing here is Thie people's mass. So the people is now twice as heavy. So that means we cannot use this equation anymore instead because the masses not constant. But we're also said that the height O r. The compression is same for both of the cases. So if we let me write that here so from there we have each over excess squared, which is equal to okay by two times. Sometimes she now, as we say, that M is not constant anymore, but excess constant so we can write aged times M, which is equal to K X, squared by two times. And she where this guy right here is constant now. So that means we can take to difference in areas and said that equal to each other. So it's one times m one will be aged two times and toe where h one is the initial height or the piece of information that we know. And everyone is also the mass of the people before we double that. So M too is basically twice off m one right. And we need to figure out aged too from there. So that's what we did here after, ah, modifying the question, we saw that m times, ages, constants. And from there we rode em one times each one being equal to him to them age too. They were solving for age to where? A 26 meters and ah masses. So one s om one is half off em too. So that's why we rode one over two over here. And from there we see that age too will be three meters, which is half off the height of the people when the compression was tiny centimeters. Thank you.

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