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A rock is thrown upward from the level ground in such a way that the maximum height of its flight is equal to its horizontal range $R$ . (a) At what angle $\theta$ is the rock thrown? (b) In terms of the original range $R$ , what is the range $R_{\max }$ the rock can attain if it is launched at the same speed but at the optimal angle for maximum range? (c) Would your answer to part (a) be different if the rock is thrown with the same speed on a different planet? Explain.

(a) $\theta = 75.96 ^ { \circ }$

(b) $R _ { \max } = 2.12 R$

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Cornell University

University of Michigan - Ann Arbor

University of Washington

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that the range equation is equaling of the initial squared multiplied by sine of tooth ada, divided by G. And we know that this is gonna be equal to the maximum heights we can save you. I final squared equals to be Why initial squared plus two times g times, Delta eight grow the other Delta Why? But Delta Y in this case would simply be h so we could say that then solving for h this would be equaling the initial squared. Or we can say the initial square times sine squared of fada divided by two G because our final velocity zero and so we're going to set this equal to the range equation. And we can immediately tell that first, these to initial velocity is gonna be, um, canceling each other out. And so sign of tooth ada is equaling sine squared of state. Uh, this would be divided, brother. This would be divided by G. This would be divided by two G. Now, sign of tooth data is simply sign of Seita Coast side of data Times two. And so we can say that given this, we can then cancel out one of these signs with one of these signs and then say that Then sign rather sign of Seita over coastline of data, which is essentially 10 of data, is equaling four. And so Seita is equaling are 10 of four. Anthea is equaling 76 degrees. So this would be the angle with us that the maximum height reached is equal to the range of the projectile. 76 degrees is our final answer for part A. Now, for part B, we can then substitute in and say our is gonna be equaling of the initial squared sine of tooth data over G. So we can say that this would be equally the initial squared. Ah, we could say then to the initial square times sign of 76 degrees co sign of 76 degrees divided by G and that our max is equally our Max would occur at Fada equaling 45 degrees 45 degrees times two is 90 degrees. Sign of many degrees is one. So this would be the initial squared over G, and we can say that our max over huh would be equaling the initial squared over G, divided by two V initial squared sine of 76 degrees co sign of 76 degrees divided by G. Of course, this is gonna cancel out, and we have that are Max over. Our is approximately equaling 2.13 So we could say for part B that r maxes equaling 2.13 times are so that would be our final answer. And that's simply moving this over. And for part C, the question is asking us if the the answer to port A would be different if the rock was thrown with the same speed on a different planet and we can say no, uh, we can say no. Fada calculation is independent of wth e acceleration due to gravity, so theta would be the same on all planets. That is the end of the solution. Thank you for watching.