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Unformatted text preview: lavallo (jhl936) Homework 05 florin (58140) 1 This printout should have 13 questions. Multiplechoice questions may continue on the next column or page find all choices before answering. 001 (part 1 of 2) 10.0 points The speed of an arrow fired from a compound bow is about 15 m / s. An archer sits astride his horse and launches an arrow into the air, elevating the bow at an angle of 43 above the horizontal and 1 . 8 m above the ground. The acceleration of gravity is 9 . 81 m / s 2 . b b b b b b b b b b b b b b b b b b b b 1 5 m / s 4 3 1 . 8 m range What is the arrows range? Assume: The ground is level. Ignore air resistance. Correct answer: 24 . 6701 m. Explanation: Let : v o = 15 m / s , = 43 , h = 1 . 8 m , and g = 9 . 81 m / s 2 . b b b b b b b b b b b b b b b b b b b b v o h range In the projectile motion, we have x = ( v o cos ) t t = x v o cos horizontally, and y = h + ( v o sin ) t 1 2 g t 2 vertically. Thus y = h + x v o sin v o cos  1 2 g x 2 v 2 o cos 2 = 0 when the arrow lands. Thus 2 v 2 o h cos 2 + 2 x v 2 o sin cos  g x 2 = 0 2 v 2 o h cos 2  x v 2 o sin2 + g x 2 = 0 x = v 2 o sin 2 radicalBig v 4 o sin 2 2 + 8 g v 2 o h cos 2 2 g . Since v 4 o sin 2 2 + 8 g v 2 o h cos 2 = (15 m / s) 4 (sin 2 86 ) + (8) (9 . 81 m / s 2 ) (15 m / s) 2 (1 . 8 m) (cos 2 43 ) = 67379 . 4 m 2 / s 2 , then x = (15 m / s) 2 (sin86 ) 2 (9 . 81 m / s 2 ) + radicalbig 67379 . 4 m 2 / s 2 2 (9 . 81 m / s 2 ) = 24 . 6701 m . 002 (part 2 of 2) 10.0 points Now assume his horse is at full gallop, moving in the same direction as he will fire the arrow, and that he elevates the bow the same way as before. What is the range of the arrow at this time if the horses speed is 44 m / s? Correct answer: 123 . 618 m. Explanation: The arrows speed relative to the ground and the angle of elevation relative to the ground are changed v x = v arrow + v archer = (15 m / s) cos 43 + 44 m / s = 54 . 9703 m / s . lavallo (jhl936) Homework 05 florin (58140) 2 v y = (15 m / s) sin 43 = 10 . 23 m / s The arrows speed relative to the ground is then given by v o = radicalBig v 2 x + v 2 y = radicalBig (54 . 9703 m / s) 2 + (10 . 23 m / s) 2 = 55 . 9141 m / s , and the angle of elevation relative to the ground is = tan 1 v y v x = tan 1 parenleftbigg 10 . 23 m / s 54 . 9703 m / s parenrightbigg = 10 . 5421 . With the new speed v and angle , the new range x can be found similarly to the previous part as x = v 2 o sin 2 2 g radicalBig v 4 o sin 2 2 + 8 g v 2 o h cos 2 2 g ....
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 Spring '10
 Florin

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