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$\bullet$ $\bullet$ $\bullet$ A ball is thrown upward with an initial velocity of15 $\mathrm{m} / \mathrm{s}$ at an angle of $60.0^{\circ}$ above the horizontal. Use energyconservation to find the ball's greatest height above theground.

$h=8.61 \mathrm{m}$

Physics 101 Mechanics

Chapter 7

Work and Energ

Physics Basics

Applying Newton's Laws

Kinetic Energy

Potential Energy

Energy Conservation

Cornell University

University of Michigan - Ann Arbor

McMaster University

Lectures

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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

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