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Carnegie Mellon University

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Problem 7 Medium Difficulty

A student stands at the edge of a cliff and throws a stone horizontally over the edge with a speed of 18.0 $\mathrm{m} / \mathrm{s}$ The cliff is 50.0 $\mathrm{m}$ above a flat, horizontal beach as shown in Figure $\mathrm{P} 3.7$ . (a) What are the coordinates of the initial position of the stone? (b) What are
the components of the initial velocity? (c) Write the equations for the $x$ - and $y$ -components of the velocity of the stone with time. (d) Write the equations for the position of the stone with time, using the coordinates in Figure P3.7. (e) How long after being released does the stone strike the beach below the cliff? (f) With what speed and angle of impact does the stone land?

Answer

(a)$x _ { 0 } = 0$
$y _ { 0 } = + 50.0 \mathrm { m }$
(b)$t _ { 0 x } = + 18.0 \mathrm { m } / \mathrm { s }$
$\nu _ { 0 y } = 0$
(c)$v _ { x } = 18.0 \mathrm { m } / \mathrm { s }$
$v _ { y } = - \left( 9.80 \mathrm { m } / \mathrm { s } ^ { 2 } \right) t$
(d)$x = ( 18.0 \mathrm { m } / \mathrm { s } ) t$
$y = 50.0 \mathrm { m } - \left( 4.90 \mathrm { m } / \mathrm { s } ^ { 2 } \right) t ^ { 2 }$
(e) $t = 3.19 \mathrm { s }$
(f)36.1$\mathrm { m } / \mathrm { s }$ at $60.1 ^ { \circ }$ below the horizontal

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Video Transcript

So for party, we can say that with the origin chosen already and figure p 3 to 3. Ah, we can say that the co ordinates of the original position of the stone we can say ex initial Why initial are gonna be equaling zero comma positive 50.0 units would of course, be meters. And for part B, we can say that um, the initial velocity of the stone in the ex direction would be equaling positive 18.0 meters per second. The initial velocity and the Y direction would be equaling zero meters per second. And we can then say for part C, um, the components of the stones velocity during its flight are of course, functions of time. And in the ex direction, there isn't any acceleration. So the, uh, the velocity in the extraction is going to stay constant. So this would be 18.0 meters per second and then for velocity. Why we can say velocity bye. Final equals velocity. Why initial plus the acceleration of the wire direction times t. Our initial velocity is, of course, zero. So our final velocity would simply be equaling negative 9.80 meters per second squared multiplied by t So these would be our two answers for part C four part D. Then we know that the coordinates of the stone during its flight would be we can say X final equals x initial, which we can just say to be zero plus the ex inertial tear plus 1/2 times a sub x t squared. We know that there isn't any acceleration in the ex direction so we can eliminate that term as well. And simply X final is gonna be equaling 18.0 meters per second multiplied by t. It's gonna be the exact same thing for the y position where we have wide final equal and why initial Plus the why initial t plus 1/2 ace of y t squared. And so here Why final is gonna be equaling 50.0 meters minus 4.90 meters per second square. So half of G multiplied by t squared. Where here velocity Why initial is zero and so these would be heir to answers for part D For a part e, Then we can find the time of the fall where Delta y is equaling the why initial T plus 1/2 times a sub y t squared and so again v Y initial zero. So we can simply say that T would be equaling the square root of two times delta y divided by the acceleration A. So this would be equaling the square root of two multiplied by negative 50 because the stone is falling 50 meters divided by negative 9.80 meters per second squared and this is giving us 3.19 seconds. This would be our answer for Part E and then four part F. We can then say that at impact velocity X final equals velocity ex initial which is really no again 18.0 meters per second. The vertical component V why final would be equaling negative 9.80 meters per second squared multiplied by 3.19 seconds. This is giving us negative 31.3 meters per second and thus the velocity the final. The magnitude of the final velocity would be equaling square root of the X component squared, plus the why component squared and this is giving us 36.1 meters per second. So this would be our magnitude and then our direction Fate up would be equaling arc. 10 of the why component the why final divided by the X final the X component. And so this is giving us here. We would have negative 60 0.1 degrees. Or we can simply say fada is equaling 60 0.1 degrees below Be positive X axis. So this would be our direction for part deep for a part f that is the end of the solution. Thank you for watching.

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