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Carnegie Mellon University

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Problem 36 Hard Difficulty

In is In a local diner, a customer slides an empty coffee cup down the counter for a refill. The cup slides off the counter and strikes the floor at distance $d$ from the base of the counter. If the height of the counter is $h,(\text { a })$ find an expression for the time $t$ it takes the cup to fall to the floor in terms of the variables $h$ and $g$ . (b) With what speed does the mug leave the counter? Answer in terms of the variables $d, g,$ and $h .(\mathrm{c})$ In the same terms, what is the speed of the cup immediately before it hits the floor? (d) In terms of $h$ and $d$ , what is the direction of the cup's velocity immediately before it hits the floor?

Answer

(a) $t = \sqrt { \frac { 2 h } { g } }$
(b) $v _ { \mathrm { x } _ { 1 } } = d \sqrt { \frac { g } { 2 h } }$
(c) $v = \sqrt { \frac { d ^ { 2 } g + 4 \sigma h ^ { 2 } } { 2 h } }$
(d) $\theta = \tan ^ { - 1 } \left( \frac { 2 h } { d } \right)$

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Video Transcript

so for here that the time the time of flight of the Coffee Cup can be Delta y equals V Y initial T plus 1/2 GT squared as it slides off the counter, it's not gonna have any initial wind velocity. And so we can say that the time would simply be equal the square root of two times the height of the counter divided by G. And so that's the time it takes for the floor to reach the ground. That would be for part a and then for part B, we could say the horizontal distance. Ah, we can say D were rather horizontal distance. Delta X equals the ex initial times t and so Ah, we can save e ex initial would be equally, uh, Delta X multiplied by the square root of G, divided by two h. So this would be our ex initial velocity in the ex direction four parts. See, Then we can I say that here the final velocity and the Y direction would be equal the initial velocity y direction plus ace of y T. And so we can say that this is gonna be zero, and so we can say that velocity why final would be equaling negative g multiplied by to H over G. And so this would be equally to negative radical to G H. And so we can solve for the final the magnitude of the final velocity. This would be equal in the square root of de radical G over to age quantity squared, plus negative radical to G H Cornered, he squared and this is equaling the square root of D squared g over to H plus two g h. This would be the final speed of the coffee cup right before it hit the floor. And this is a counting. This is the magnitude of the velocity. So this is accounting for the X component and the white component. And then for Port de we can simply say that arc, that tan Dr Death ada would be arc 10 of the y component of velocity negative radical to G H, divided by the X component of the velocity de multiplied by radical G over to H. And so we find that data is equaling arc 10. So the angle at which the coffee cup hits the counter or the direction of the final velocity of the coffee cup right before it hits the ground. Not the counter, but rather the ground. This would be equally arc 10 of negative to H over G. So this would be our final answer for Part D. That is the end of the solution. Thank you for watching.

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