# hw4 - lavallo(jhl936 – Homework 05 – florin –(58140 1...

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Unformatted text preview: lavallo (jhl936) – Homework 05 – florin – (58140) 1 This print-out should have 13 questions. Multiple-choice 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 arrow’s range? Assume: The ground is level. Ignore air resistance. Correct answer: 24 . 6701 m. Explanation: Let : v o = 15 m / s , θ = 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 θ h range In the projectile motion, we have x = ( v o cos θ ) t t = x v o cos θ horizontally, and y = h + ( v o sin θ ) t- 1 2 g t 2 vertically. Thus y = h + x v o sin θ v o cos θ- 1 2 g x 2 v 2 o cos 2 θ = 0 when the arrow lands. Thus 2 v 2 o h cos 2 θ + 2 x v 2 o sin θ cos θ- g x 2 = 0- 2 v 2 o h cos 2 θ- x v 2 o sin2 θ + g x 2 = 0 x = v 2 o sin 2 θ ± radicalBig v 4 o sin 2 2 θ + 8 g v 2 o h cos 2 θ 2 g . Since v 4 o sin 2 2 θ + 8 g v 2 o h cos 2 θ = (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 horse’s speed is 44 m / s? Correct answer: 123 . 618 m. Explanation: The arrow’s 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 arrow’s 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 θ ′ = 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 θ ′ , the new range x ′ can be found similarly to the previous part as x ′ = v ′ 2 o sin 2 θ ′ 2 g ± radicalBig v ′ 4 o sin 2 2 θ ′ + 8 g v ′ 2 o h cos 2 θ ′ 2 g ....
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## This note was uploaded on 04/09/2010 for the course PHY 58150 taught by Professor Florin during the Spring '10 term at University of Texas.

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hw4 - lavallo(jhl936 – Homework 05 – florin –(58140 1...

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