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Carnegie Mellon University

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Problem 55 Hard Difficulty

A golf ball with an initial speed of 50.0 $\mathrm{m} / \mathrm{s}$ lands exactly 240 $\mathrm{m}$ downrange on a level course. (a) Neglecting air friction, what two projection angles would achieve this result? (b) What is the maximum height reached by the ball, using the two angles determined in part (a)?

Answer

(a) $\theta _ { 1 } = 35.1 ^ { \circ }$ and $\theta _ { 2 } = 54.9 ^ { \circ }$
(b) $H _ { 1 } = 42.17 \mathrm { m }$ and $H _ { 2 } = 85.38 \mathrm { m }$

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Video Transcript

so we know for part A that the range is equaling essentially Delta X. And this would be equaling the initial squared sine of tooth ada, divided by G. And so solving weaken safe ADA is gonna be equaling G Delta X. Or rather, you could say 1/2 multiplied by arc sine Uh huh. G Delta X divided by V initial squared. And so we can then find data Saito would be equaling 1/2 want supplied by arc sine of 9.80 meters per second squared, multiplied by 240 meters, divided by 50.0 meters per second quantity squared. And this is giving us 35.1 degrees and 54.9 degrees. So of course they're going to be compliments of one another. These will be our two answers for part a No. Four part B. We want to find the maximum height and you know the maximum height, uh, at max height. We can say that the velocity in the UAE direction final is equaling zero meters per second or at that point, and so we can firsts find the elapsed time and so we can say the time at the peak would essentially be equaling the wife final at the peak minus B. Why initial This would be divided by the acceleration in the UAE direction. And so this would simply be equaling negative v initial sign of state, uh, divided by negative G. Or we can simply say the initial sign of Fada divided by G. And so the y coordinate of the ball. At this time we could say why, Max, this would be equaling v Y initial t plus 1/2 times the acceleration in the wind direction t squared. And this is gonna become the initial sign of fada multiplied by V initial sign of fada over D G minus G over too. Multiplied by V initial sign squared the fate of the initial squared sine squared of theta over G squared. And so we can then see that why sub Max or the maximum height is gonna be equaling the initial squared sine of tooth Ada Oh, it's all right Sine squared a fade on my apologies divided by two G. And so we can then say after plugging in for both of our angles, we have why, Max, this would be for part being. Why, Max, you could say someone would be equaling 50 meters per second quantity squared sine squared of 35.1 degrees. This would be divided by two times 9.80 meters per second squared. And this is giving us 42.2 meters. And then why Max? Some two is equally again. 50 meters per second quantity squared, sine squared of here it would be 54 0.9 degrees again divided by two times 9.80 meters per second squared. And this is giving us 85.4 meters. So these would be our two answers for part B, using both of our angles from part. Eh? That is that is the end of the solution. Thank you for watching.

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