08_P55InstructorSolution

# 08_P55InstructorSolution - 8.55 Model Visualize Solve...

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8.55. Model: Assume the particle model and apply the constant-acceleration kinematic equations. Visualize: Solve: (a) Newton’s second law for the projectile is ( ) net wind x x F F ma = − = G x F a m = where wind F is shortened to F . For the y -motion: ( ) ( ) 2 1 1 0 0 1 0 1 0 2 y y y y v t t a t t = + + ( ) 2 1 0 1 1 2 0 m 0 m sin v t gt θ = + 1 0 s t = and 0 1 2 sin v t g θ = Using the above expression for 1 t and defining the range as R we get from the x motion: ( ) ( ) 2 1 1 0 0 1 0 1 0 2 x x x x v t t a t t = + + 2 1 1 0 0 1 1 2 x F x x R v t t m = = + ( ) 2 0 0 0 2 sin 2 sin cos 2 v F v v g m g θ θ θ = 2 2 2 0 0 2 2 2 cos sin sin v v F g mg θ θ θ = We will now maximize R as a function of θ by setting the derivative equal to 0: ( ) 2 2 2 2 0 0 2 2 2 cos sin 2sin cos 0 dR v Fv d g mg θ θ θ θ θ = = 2 2 2 0 2 2 0 2 cos sin cos2 sin2 2 Fv g mg v θ θ θ θ = = ⎠⎝ tan2 mg F θ = Thus the angle for maximum range is ( ) 1 1 2 tan / mg F θ = (b) We have ( ) ( ) 2 0.50 kg 9.8 m/s 8.167

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