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Carnegie Mellon University

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Problem 34 Easy Difficulty

You can use any coordinate system you like to solve a projectile motion problem. To demonstrate the truth of this statement, consider a ball thrown off the top of a building with a velocity $\overrightarrow{\mathbf{v}}$ at an angle $\theta$ with respect to the horizontal. Let the building be 50.0 $\mathrm{m}$ tall, the initial horizontal velocity be 9.00 $\mathrm{m} / \mathrm{s}$ , and the initial vertical velocity be 12.0 $\mathrm{m} / \mathrm{s}$ . Choose your coordinates such that the positive $y$ -axis is upward, the $x$ -axis is to the right, and the origin is at the point where the ball is released. (a) With these choices, find the ball's maximum height above the ground and the time it takes to reach the maximum height. (b) Repeat your calculations choosing the origin at the base of the building.

Answer

(a) $h _ { \max } = 57.34 \mathrm { m }$ and $t = 1.22 \mathrm { s }$
(b) $h _ { \max } = 7.34 \mathrm { m }$ and $t = 1.22 \mathrm { s }$

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Video Transcript

{'transcript': "turn a the initial coordinates air simply zero comma y initial four parts to be then velocity ex initial would be equally velocity official co sign of data. This'd be equaling 8.0 meters per second times co sign, uh, negative 20 degrees and velocity. Ex initial is equaling 7.52 meters per second. Velocity wind shows equaling velocity. Initial sign of data. This is equaling 8.0 meters per second sign. Uh, negative 20 0.0 degrees and velocity wind ish Eliza equaling negative 2.74 meters per second. So these would be heir to answers for part B and for part C. Then we can say that the exposition we know the the change of displacement is gonna be equaling velocity accident show plus times t rather because there isn't any acceleration in the ex direction. And so this is simply good of equaling 7.52 meters per second times t We know that for the y direction this would be the Y initial t plus 1/2 gt squared and here we're choosing downwards to be negative. So this would be equally negative 2.74 meters per second times T minus Rather would be plus 1/2 multiplied by negative 9.80 meters per second squared times t squared and so delta y four. Better yet, we could say why final is going to be equally Why initial plus my initial minus 2.74 meters per second T minus 4.90 meters per second squared t squared. And so this would be the X and Y components, the final X and y components of the ball after Time t So we have this in this and at this point would be for part C for party. Then we went to the plug ID for three seconds essentially so this would be out or other. Better yet, X final equal 7.52 meters per second multiplied by 3.0 seconds. This is equally 22.6 meters. Why final would equal why initial which is we know to be zero. This would be a plus negative 2.74 meters per second times 3.0 seconds minus 4.90 years per second. Squared well supplied by 3.0 seconds quantity squared and we have our wife final equaling. Um, rather my apologies. This this is correct. However, here it's actually by final. That would be zero, because we're choosing again downwards to be negative. So our initial height would be equaling positive 52.3 meters. This would be our final answer for the initial vertical height. Again, we chose downwards to the negative. So that means that our wife final is gonna be zero rather than a Why initial. My apologies are why initial is what we're trying to find and so that would be 52.3 meters. So I would be your answer for Park D and this would be a X Final four part D. This is your eyes for part D. This would be a prince for Part E and then we have part off. So this would be the final velocity of the ball in the wind direction. Velocity Y final squared equals velocity wind initials where plus two times the acceleration due to gravity times out, the why commedia would be why the y final equals the square root of negative 2.74 meters per second quantity squared plus two times negative 9.80 meters per second squared times negative 10.0 meters and so velocity y final is equaling negative 14.3 meters per second t Someone is an equaling velocity wife final minus velocity Why initial divided by acceleration? Why direction? This is equally negative 14.3 minus negative 2.74 again that units here our meters per second divided by negative 9.80 meters per second squared and we find that the required elapsed time t someone is equally 1.18 seconds. This would be a final answer for part F. That is the end of the solution. Thank you for watching."}

Carnegie Mellon University
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