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Problem 1.16 - Problem 1.16[Difficulty 3 Given Long bow at...

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Problem 1.16 [Difficulty: 3] Given: Long bow at range, R = 100 m. Maximum height of arrow is h = 10 m. Neglect air resistance. Find: Estimate of (a) speed, and (b) angle, of arrow leaving the bow. Plot: (a) release speed, and (b) angle, as a function of h Solution: Let V u i v j V i j) 0 0 0 = + = + 0 0 0 ± ± (cos ± sin ± θ θ Σ F m mg y dv dt = = − , so v = v 0 – gt, and t f = 2t v=0 = 2v 0 /g Also, mv dv dy mg, v dv g dy, 0 v 2 gh 0 2 = − = − = − Thus h v 2g 0 2 = (1) Σ F m du dt 0, so u u const, and R u t 2u v g 0 0 f 0 0 x = = = = = = (2) From Eq. 1: v 2gh 0 2 = (3) From Eq. 2: u gR 2v gR 2 2gh u gR 8h 0 0 0 2 2 = = = Then 2 1 2 0 2 2 0 2 0 2 0 8 2 and 2 8 + = + = + = h gR gh V gh h gR v u V (4) s m 7 . 37 m 10 1 m 100 s m 8 81 . 9 m 10 s m 81 . 9 2 2 1 2 2 2 2 0 = × × + × × = V From Eq. 3:
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