Question-169969

Question Number 169969 by Beginner last updated on 13/May/22

Answered by mr W last updated on 13/May/22

Commented by ajfour last updated on 13/May/22

u = \sqrt{2 g h}

(u \sin θ) t + \frac{(g \sin θ) t^{2}}{2} = s

(u \cos θ) t - \frac{(g \cos θ) t^{2}}{2} = 0

\Rightarrow t = \frac{2 u}{g}

s = \frac{2 u^{2} \sin θ}{g} + \frac{2 u^{2} \sin θ}{g}

s = \frac{4 (2 g h) \sin θ}{g}

s = 8 h \sin θ

Commented by mr W last updated on 13/May/22

g r e a t s i r!

Commented by learner22 last updated on 13/May/22

b r a v o

Commented by mr W last updated on 13/May/22

Commented by mr W last updated on 13/May/22

a s s u m e t h a t t h e b a l l h i t s t h e p l a n e

a g a i n, t h e n

w i t h λ = \frac{a}{h} = \frac{l \sin θ}{h} a n d ξ = \frac{l}{h} w e h a v e

\cos 2 θ + \sqrt{\cos^{2} 2 θ + ξ \sin θ} = \frac{ξ \sin θ}{2 (1 - \cos 2 θ)}

\sqrt{\cos^{2} 2 θ + ξ \sin θ} = \frac{ξ \sin θ}{2 (1 - \cos 2 θ)} - \cos 2 θ

\Rightarrow ξ = 8 \sin θ

Commented by mr W last updated on 13/May/22

s a y i m p a c t p o i n t B i s a a b o v e t h e

g r o u n d .

s p e e d a t p o i n t B :

v = \sqrt{2 g h}

u p o n p o i n t B :

x = v \sin 2 θ t

y = a + v \cos 2 θ t - \frac{{g t}^{2}}{2}

y = a + v \cos 2 θ \times \frac{x}{v \sin 2 θ} - \frac{g}{2} \times {(\frac{x}{v \sin 2 θ})}^{2}

y = a + \frac{x}{\tan 2 θ} - \frac{{g x}^{2}}{2 v^{2} \sin^{2} 2 θ}

\Rightarrow y = a + \frac{x}{\tan 2 θ} - \frac{x^{2}}{4 h \sin^{2} 2 θ}

a t y = 0 :

a + \frac{x}{\tan 2 θ} - \frac{x^{2}}{4 h \sin^{2} 2 θ} = 0

x^{2} - \frac{4 h \sin^{2} 2 θ x}{\tan 2 θ} - 4 a h \sin^{2} 2 θ = 0

x = \frac{2 h \sin^{2} 2 θ}{\tan 2 θ} + 2 \sin 2 θ \sqrt{h (h \cos^{2} 2 θ + a)}

s u c h t h a t t h e b a l l h i t s t h e p l a n e a g a i n,

x = \frac{2 h \sin^{2} 2 θ}{\tan 2 θ} + 2 \sin 2 θ \sqrt{h (h \cos^{2} 2 θ + a)} ⩽ b = \frac{a}{\tan θ}

\frac{2 \sin^{2} 2 θ}{\tan 2 θ} + 2 \sin 2 θ \sqrt{\cos^{2} 2 θ + \frac{a}{h}} ⩽ \frac{a}{h \tan θ}

l e t λ = \frac{a}{h}

\Rightarrow \cos 2 θ + \sqrt{\cos^{2} 2 θ + λ} ⩽ \frac{λ}{2 (1 - \cos 2 θ)}

e x a m p l e : θ = 30 °

λ = 2.2873

i . e . w h e n \frac{a}{h} ⩾ 2.2873, t h e b a l l w i l l

h i t t h e p l a n e a g a i n a f t e r t h e i m p a c t .

Commented by Tawa11 last updated on 14/May/22

Great sir

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