mt1_review_saturday_soln - Stefan's Review Problems for...

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Stefan’s Review Problems for Midterm 1: Solutions 1 Juggling I like to juggle. But I’m not very good. It takes me 1 . 2 s from the time I catch a ball until I throw it back into the air (after which I’m ready to catch another). a. What initial vertical speed must I give each ball if I want to juggle 5 balls? b. How many balls can I juggle if I toss each ball to a height of 2 m? a. If I’m successfully juggling 5 balls, it means that in the time it takes a single ball to leave my hand, rise through the air, and come back down to my hand, I’ve managed to toss 4 other balls. That means each ball must spend at least 4 . 8 s in the air. The question asks for vertical velocity. We know that the time t each ball spends in the air is determined by its vertical velocity according to the formula 0 = y = v 0 t - 1 2 gt 2 . So solving for t I get t = 2 v 0 g . This must be greater than 4 . 8 s , and so we get the final answer v 0 > 24 m / s . b. In this case, throwing each ball up to a certain height will determine how long they spend in the air, which will determine how long I have to catch and throw the other balls. So how long does it take a ball to rise to 2 m and then fall back down? One way to figure this out is to find the initial velocity necessary to make the ball rise 2 m and then use the method from the previous part to find the time. We can find the initial velocity by using the time-independent kinematic formula: 0 2 = v 2 0 - 2 g ( 2 m ) , 1
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v 0 = 2 10 m / s . Now we find that t = 4 10 s = 1 . 26 s . Since it takes me 1 . 2 s to toss a ball, I only have time to toss one more ball while the first one is in the air. So the number of balls is 2 . 2 Uphill Battle A projectile is fired from the base of a hill that rises with angle θ above the horizontal. The projectile is fired with muzzle speed v 0 . a. If the projectile is fired at an angle φ > θ above the horizontal, how far up the hill will it land? b. Show that this expression reduces to what you would expect for θ = 0 . c. How would you find the angle φ that gave the maximum range? (You don’t have to explicitly solve for it.) a. Say the projectile is fired a distance D up the hill (so we’re trying to solve for D .
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