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Uara mechann energy consewvation theorem de"ermine tr Iighest Tasion ball thai is launched upward with intial veboly 15m/5 Using kinematics equation calculate ...

Question

Uara mechann energy consewvation theorem de"ermine tr Iighest Tasion ball thai is launched upward with intial veboly 15m/5 Using kinematics equation calculate tne highest positon [ the ball can feach assuming thc sme Initia velocity (15m/s) of the ball Do Ihe to approaches Din] abcntne sme Toeull?

Uara mechann energy consewvation theorem de"ermine tr Iighest Tasion ball thai is launched upward with intial veboly 15m/5 Using kinematics equation calculate tne highest positon [ the ball can feach assuming thc sme Initia velocity (15m/s) of the ball Do Ihe to approaches Din] abcntne sme Toeull?



Answers

A ball is thrown upward with an initial velocity of 15 m/s at an angle of 60.0$^\circ$ above the horizontal. Use energy conservation to find the ball's greatest height above the ground.

So that's first draw this an area here and then we can talk about how to solve the problem. So the ball is thrown at 60 degree angles from the horizontal and then it drops comes back to the ground. So let's say this is thie maximum height eight, which we need to figure out. And the direction of the velocity is this way, which is making 60 degree angles from the horizontal now at the maximum point, the velocity. So when we have a projectile, the velocities always tangential toe. At each point, the velocities always standing, shelter the projectile. So at this point, the velocity will be directing Ah, parallel e to the ground. So that means if the initial direction, the velocity 60 degrees from the horizontal, we can shift this over here and make this angle so this velocity will be course I in 60 degree off the initial velocity. So if we call the initial velocity as it's a B e I. So the velocity at this point, let's call this point as if the velocity at this point is the final. So we'll have the final, which is equal to the initial velocity times co sign 60 degrees. All right, so now we can talk about a few more things, like for the final high. If we consider this as our origin, the ground is our origin. So for why access this point or this point will be zero. So let's call it why I with zero and at the top we call it y f. Then why f will be the maximum height h. Now we can apply the conservation off energy around this point and this point. So if we do so will have key Final plus you final, which is equal to K initial. Plus you initial where keys, the kind of energy and use the potential energy. Now I can immediately see that why I is zero. So that means we can ignore the potential energy at ground. And ah, so we'll have for kinetic energy will have half envy. I squared. So let's write that down. And for the final part, we'll have m g h for the potential energy and for the kinetic energy will have half em. He f squared so we'll have half and VF squared. So let me write down the question one more time on this page, so we have half off M V I squared, which is equal to M times C times, age plus half M V F squared where V if is the clothes and six degrees. So let's evaluate that for a moment. So we have assigned six degrees and via is 15 meter per second and co sign 60 is half. So that gives a 7.5 meters per second Sophie, even us all for the hide age. We'll have h equal to the nation square so we can get rid off this m terms. So yeah, that's gonna be V. I squared minus 7.5. Oops, meter per second. Then there's a square term. We divide that by two times nine find eight Meena per second squared. So we know that via is 15 meters per second and square it Then we have seven point 5,000,000 per second square it We divide that by two times 9.8 meters per second squared. Using that, we find H as 8.6 mirror. So that's the maximum hide the ball will reach. Thank you

All right. So this is just the basic kid in my next question, right, we have the Thea ball being a meter above the ground and we have the acceleration being just gravity negative or 9.81 meters per second squared downwards. And the initial velocity is going to be 19.6 meters per second with an angle of 30 degrees above horizontal. So what we have to do is figure out what is gonna be the height of the ball when it starts dipping again. Right? What is the peak? So in order to do that, well, we all have. All we have to do is look at the vertical sort of aspect of this question. And so let's figure out first the initial velocity, vertical velocity of the ball, we can split the initial velocity here into the horizontal and vertical components, and we can find out that v not. Why? While that's just gonna be able to 19.6 times, uh, signed 30 and that he's going to be equal to 9.8 meter second. All right, cool. So we have 9.8 meters per second. Now that we have this we can use this formula inside here to ah, do what we need. So we're really trying to find the point where the Y velocities equals zero right? Because when it reaches its peak, it's sort of, you know, temporarily has ah, velocity of zero. And so we can set the final velocity here equal to zero. And so we know the initial velocity. We know the acceleration and then we can solve for the adult X or, you know, the peak. And so let's do that. So we have reraise a formal to make it. Delta X is going to be equal to B squared minus fee not scored, divided by two a. Okay, now we don't have the score just gonna be equal to zero. So let's ignore that. And since ah, you know, the acceleration is technically negative. 9.81 meters per second squared. Well, just get this negative. Basically, we have, uh, 9.8 meters per second squared, divided by two times 9.81 user per second squared, and that will give us an answer of four point nine meters. And since the balls already one meter above the ground, the height of the ball at its peak is gonna be 4.9 meters plus one. So that's equal to 5.9 meters, right, Because we have to consider this peak. So there you go. There's the first answer. Next, we have to calculate the speed of the ball. Um, when it has reached to speak well, we know that the velocity vertical velocity is going to be zero right, because that's sort of what happens when it reaches to speak. So all we have to worry about now is the horizontal muzzle velocity. While the horizontal velocity doesn't change, there's no horizontal acceleration going on here. So the horizontal accelerate velocity is just gonna be the same thing as the initial horizontal velocity that is just gonna be equal to 19.6 music per second. Times Co signed three degrees and that, my friends, is just equal to 16 or 17 music. So that's your answer

Yeah, At the top, V equals zero. So kind of like energy at the top equals zero. Now, initial kinetic energy is half AM V I square and Connecticut a jet The midway will be half G. I equals 1/4 and V I squared and momentum Midway is M v. I equals m squared V. I squared over m equals true for four, right? Hardcore No, let's back up for a second. So kinetic energy made way is m v m equals and we will have Can you take energy? Midway equals p m squared over to em So we have be midway equals quelled off to em I am equals to him Times 1/4 m v I squared equals on that M v I divided by a squared off to on putting the numbers in This will come out to be 1.6 kilogram meter per second.

Okay in this problem, we've got a person with a ball and we know that the ball has a massive 100 grams. I'm not even gonna write that down and throws a ball straight up and it comes straight down. I'm drying it a little bit to the right just to show the up and down. But it's really going straight up and then straight down. And the question is, what's the minimum speed required in order to get to a height of 10 meters, assuming we released of all 1.5 meters from the ground when we release. So this is a conservation of energy problem. The conservation of energy equation is the initial energy is equal to the final energy. When I'm at my starting point, I've got two energies. I've got my kinetic energy, which is 1/2 m times vo squared. I've also got I'm gonna go ahead and choose my arbitrary zero as the ground because I'm trying to figure out the height of 10 meters measured from the ground that's given to me. And so it's not an equal sign. So that is equal to plus my potential energy, which is the mass times g times, my initial height, which is at 1.5. And when I get to the top right here, by definition, my velocity is going to be zero because I wanted That's the maximum height right at the point where it stops moving upward. And so all I have is mg times that final height have 10 meters. First thing I wanna do is cancel the masses. That's why he didn't write it down in the first place. Second, I'm gonna multiply everything through by two. So Vo Squared is equal to two g ties my original height. Actually, I take that back times this one times to this, I'm gonna move over to the right side. So I've got minus two g times. My original hate. So my video was going to be equal to the square root of two times g. Time is my delta H. And sometimes some people like to think of potential energy in terms of Delta H. And that's the Delta H right there. I can plug those numbers in two 9.8 and the difference in height is 8.5. I pulled out my calculator. I'm going to have to times 9.8 times 8.5. That's 166.6. Take the square root of it, then they need a speed of 12.9 meters per second in order to just barely reached that. That hype. The next question is, what's the speed? We're gonna call it V two? What's the speed just before the ball hits the ground? And there's two different ways that you can solve that you can either solve it from the original point or solve it from that maximum point that we just, uh, wait, going to say that we know that airspeed zero there and it drops 10 meters. I'm gonna go ahead. And just because it's conservation energy get used to conservation of energy. I'm gonna do it from this initial point. And so I'm going to say EI is equal to e f. I've got my exact same starting conditions 1/2 M. V o squared, plus M g h o. And then when I get down to the bottom to this point right there, I have zero potential energy by definition because I'm at the ground and I arbitrarily chose my zero to be the ground. And so whenever I'm at that point about zero potential energy. But I do have kinetic energy. 1/2 him b squared. And that's that V two that I talked about again going to cancel the EMS again. I'm gonna multiply everything by two to hit for the fractions. Vo squared plus to G times. My original height is equal to V squared, noticed by the way, that this is just one of the kids cant o Matic equations. My velocity squared is equal to my initial velocity squared plus two times my acceleration in this case make celebrations g times the delta H. And this time it's the I have to drop a height of 1.5 meters and so that's what's going on there. I'm going to go ahead and plug those in V squared is equal to no. Get rid of Squared V is equal to the square root of my vo squared. I just calculated that is 12.9 plus two times G, which is 9.8 times my original height, which is 1.5 Gonna plug those into my calculator. I'm gonna square get my 1 66.6 back. I have got to times 9.8 times 1.5. That's gonna be the square root of 196 which you may not need a calculator for. To know that that answer is V two is equal to 14. Meat is presented, and that is how you solve both of those firms.


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