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Question Number 72803 by mr W last updated on 03/Nov/19

Commented by mr W last updated on 03/Nov/19

$a b a l l i s t h r o w n f r o m p o i n t A w i t h a$ $s p e e d u . a f t e r s t r i k i n g a p o i n t o n t h e$ $i n c l i n e d p l a n e, i t r e t u r n s b a c k t o i t s$ $o r i g i n a l p o s i t i o n . i f t h e c o e f f i c i e n t o f$ $r e s t i t u t i o n i s e, h o w l a r g e m u s t u b e$ $s u c h t h a t t h e m o t i o n o f t h e b a l l a s$ $d e s c r i b e d i s p o s s i b l e .$
	Answered by ajfour last updated on 03/Nov/19

	Commented by ajfour last updated on 04/Nov/19

	$I t g e t s c o m p l i c a t e d S i r, i t r i e d$ $b u t p o s t e d a n d d e l e t e d . .$
	Commented by mr W last updated on 04/Nov/19

	$t h a n k s f o r t r y i n g s i r!$ $i t i s r e a l l y c o m p l i c a t e d .$
	Commented by ajfour last updated on 04/Nov/19
	$let tan θ=s, tan α=m, tan δ=n H=sD−((gD^2 (1+s^2 ))/(2u^2 )) ....(i) −H=nD−((gD^2 (1+n^2 ))/(2v_2 ^2 )) ...(ii) mD=h+H ...(iii) v_1 ^2 =u^2 −2gH ...(iv) v_2 cos φ=ev_1 cos ϕ v_2 sin φ=v_1 sin ϕ unknowns u, m, n, D, H, ϕ, φ v_2 ^2 =v_1 ^2 (e^2 cos^2 ϕ+sin^2 ϕ) ⇒ v_2 ^2 =(u^2 −2gH)(e^2 cos^2 ϕ+sin^2 ϕ) etan φ=tan ϕ tan δ=n=tan (90°−α−φ) ⇒ n=((1−tan φ)/(m+tan φ))=((e−tan ϕ)/(em+tan ϕ)) tan (90°−α+ϕ)=((√(u^2 sin^2 θ−2gH))/(ucos θ)) ⇒ ((1+mtan ϕ)/(m−tan ϕ))=(√(s^2 −((2gH(1+s^2 ))/u^2 ))) = f .....(v) 1+mtan ϕ=mf−ftan ϕ ⇒ tan ϕ=((mf−1)/(m+f)) ....(vi) n= ((e−((mf−1)/(m+f)))/(em+((mf−1)/(m+f))))=((em+ef−mf+1)/(em^2 +emf+mf−1)) ....(vii) from (ii) which is −H=nD−((gD^2 (1+n^2 ))/(2v_2 ^2 )) __________________________ −H=nD−((gD^2 (1+n^2 ){(mf−1)^2 +(m+f)^2 })/(2(u^2 −2gH)[e^2 +(mf−1)^2 ])) −H=(((em+ef−mf+1)/(em^2 +emf+mf−1)))D−((gD^2 {1+(((em+ef−mf+1)/(em^2 +emf+mf−1)))^2 }{(mf−1)^2 +(m+f)^2 })/(2(u^2 −2gH)[e^2 +(mf−1)^2 ])) ...(1) where f=(√(s^2 −((2gH(1+s^2 ))/u^2 ))) mD=h+H (2) ...(iii) H=sD−((gD^2 (1+s^2 ))/(2u^2 )) (3) ....(i) Now unknowns are u, s, H, D From (1), (2), (3) we obtain a correlation in u and s=tan θ ...........■$
	$l e t \tan θ = s, \tan α = m, \tan δ = n$ $H = s D - \frac{{g D}^{2} (1 + s^{2})}{2 u^{2}} . . . . (i)$ $- H = n D - \frac{{g D}^{2} (1 + n^{2})}{2 v_{2}^{2}} . . . (i i)$ $m D = h + H . . . (i i i)$ $v_{1}^{2} = u^{2} - 2 g H . . . (i v)$ $v_{2} \cos ϕ = {e v}_{1} \cos φ$ $v_{2} \sin ϕ = v_{1} \sin φ$ $u n k n o w n s u, m, n, D, H, φ, ϕ$ $v_{2}^{2} = v_{1}^{2} (e^{2} \cos^{2} φ + \sin^{2} φ)$ $\Rightarrow v_{2}^{2} = (u^{2} - 2 g H) (e^{2} \cos^{2} φ + \sin^{2} φ)$ $e \tan ϕ = \tan φ$ $\tan δ = n = \tan (90 ° - α - ϕ)$ $\Rightarrow n = \frac{1 - \tan ϕ}{m + \tan ϕ} = \frac{e - \tan φ}{e m + \tan φ}$ $\tan (90 ° - α + φ) = \frac{\sqrt{u^{2} \sin^{2} θ - 2 g H}}{u \cos θ}$ $\Rightarrow \frac{1 + m \tan φ}{m - \tan φ} = \sqrt{s^{2} - \frac{2 g H (1 + s^{2})}{u^{2}}}$ $= f . . . . . (v)$ $1 + m \tan φ = m f - f \tan φ$ $\Rightarrow \tan φ = \frac{m f - 1}{m + f} . . . . (v i)$ $n = \frac{e - \frac{m f - 1}{m + f}}{e m + \frac{m f - 1}{m + f}} = \frac{e m + e f - m f + 1}{{e m}^{2} + e m f + m f - 1}$ $. . . . (v i i)$ $f r o m (i i) w h i c h i s$ $- H = n D - \frac{{g D}^{2} (1 + n^{2})}{2 v_{2}^{2}}$ $__________________________$ $- H = n D - \frac{{g D}^{2} (1 + n^{2}) {{(m f - 1)}^{2} + {(m + f)}^{2}}}{2 (u^{2} - 2 g H) [e^{2} + {(m f - 1)}^{2}]}$ $- H = (\frac{e m + e f - m f + 1}{{e m}^{2} + e m f + m f - 1}) D - \frac{{g D}^{2} {1 + {(\frac{e m + e f - m f + 1}{{e m}^{2} + e m f + m f - 1})}^{2}} {{(m f - 1)}^{2} + {(m + f)}^{2}}}{2 (u^{2} - 2 g H) [e^{2} + {(m f - 1)}^{2}]}$ $. . . (1)$ $w h e r e f = \sqrt{s^{2} - \frac{2 g H (1 + s^{2})}{u^{2}}}$ $m D = h + H (2) . . . (i i i)$ $H = s D - \frac{{g D}^{2} (1 + s^{2})}{2 u^{2}} (3) . . . . (i)$ $N o w u n k n o w n s a r e u, s, H, D$ $F r o m (1), (2), (3) w e o b t a i n$ $a c o r r e l a t i o n i n u a n d s = \tan θ$ $. . . . . . . . . . . ◼$
	Commented by mr W last updated on 04/Nov/19

	$c o r r e c t w a y s i r!$
	Answered by mr W last updated on 04/Nov/19

	Commented by mr W last updated on 04/Nov/19
	$AB=s v_1 sin ϕ=v_2 sin φ ev_1 cos ϕ=v_2 cos φ ((tan ϕ)/e)=tan φ ⇒sin φ=((tan ϕ)/(√(e^2 +tan^2 ϕ)))=(1/(√(1+(e^2 /(tan^2 ϕ))))) ⇒φ=tan^(−1) ((tan ϕ)/e) ⇒v_2 =((sin ϕ)/(sin φ))v=(√(sin^2 ϕ+e^2 cos^2 ϕ)) v t=((s cos α)/(v cos ((π/2)−α+ϕ)))=((s cos α)/(v sin (α−ϕ))) −s sin α+h=v cos (α−ϕ) t−((gt^2 )/2) ((gs^2 cos^2 α)/(2v^2 ))[1+(1/(tan^2 (α−ϕ)))]−[tan α+(1/(tan (α−ϕ)))]s cos α+h=0 ⇒((s cos α)/(2h))=(1/(((2gh)/v^2 )[1+(1/(tan^2 (α−ϕ)))])){tan α+(1/(tan (α−ϕ)))+(√([tan α+(1/(tan (α−ϕ)))]^2 −((2gh)/v^2 )[1+(1/(tan^2 (α−ϕ)))]))} with V=((2gh)/v^2 ) ⇒((s cos α)/(2h))=(1/(V[1+(1/(tan^2 (α−ϕ)))])){tan α+(1/(tan (α−ϕ)))+(√([tan α+(1/(tan (α−ϕ)))]^2 −V[1+(1/(tan^2 (α−ϕ)))]))} t=((s cos α)/(v_2 cos ((π/2)−α−φ)))=((s cos α)/(v sin (α+φ)(√(sin^2 ϕ+e^2 cos^2 ϕ)))) −s sin α+h=v_2 cos (α+φ) t−((gt^2 )/2) −s sin α+h=((s cos α)/(tan (α+φ)))−((gs^2 cos^2 α)/(2v^2 (sin^2 ϕ+e^2 cos^2 ϕ)))[1+(1/(tan^2 (α+φ)))] ((gs^2 cos^2 α)/(2v^2 (sin^2 ϕ+e^2 cos^2 ϕ)))[1+(1/(tan^2 (α+φ)))]−[tan α+(1/(tan (α+φ)))]s cos α+h=0 ⇒((s cos α)/(2h))=(1/(((2gh)/(v^2 (sin^2 ϕ+e^2 cos^2 ϕ)))[1+(1/(tan^2 (α+tan^(−1) ((tan ϕ)/e))))])){tan α+(1/(tan (α+tan^(−1) ((tan ϕ)/e))))+(√([tan α+(1/(tan (α+tan^(−1) ((tan ϕ)/e))))]^2 −((2gh)/(v^2 (sin^2 ϕ+e^2 cos^2 ϕ)))[1+(1/(tan^2 (α+tan^(−1) ((tan ϕ)/e))))]))} ⇒((s cos α)/(2h))=(1/((V/((sin^2 ϕ+e^2 cos^2 ϕ)))[1+(1/(tan^2 (α+tan^(−1) ((tan ϕ)/e))))])){tan α+(1/(tan (α+tan^(−1) ((tan ϕ)/e))))+(√([tan α+(1/(tan (α+tan^(−1) ((tan ϕ)/e))))]^2 −(V/((sin^2 ϕ+e^2 cos^2 ϕ)))[1+(1/(tan^2 (α+tan^(−1) ((tan ϕ)/e))))]))} (1/((V/((sin^2 ϕ+e^2 cos^2 ϕ)))[1+(1/(tan^2 (α+tan^(−1) ((tan ϕ)/e))))])){tan α+(1/(tan (α+tan^(−1) ((tan ϕ)/e))))+(√([tan α+(1/(tan (α+tan^(−1) ((tan ϕ)/e))))]^2 −(V/((sin^2 ϕ+e^2 cos^2 ϕ)))[1+(1/(tan^2 (α+tan^(−1) ((tan ϕ)/e))))]))}=(1/(V[1+(1/(tan^2 (α−ϕ)))])){tan α+(1/(tan (α−ϕ)))+(√([tan α+(1/(tan (α−ϕ)))]^2 −V[1+(1/(tan^2 (α−ϕ)))]))} ⇒((tan α+(1/(tan (α−ϕ)))+(√([tan α+(1/(tan (α−ϕ)))]^2 −V[1+(1/(tan^2 (α−ϕ)))])))/(tan α+(1/(tan (α+tan^(−1) ((tan ϕ)/e))))+(√([tan α+(1/(tan (α+tan^(−1) ((tan ϕ)/e))))]^2 −(V/((sin^2 ϕ+e^2 cos^2 ϕ)))[1+(1/(tan^2 (α+tan^(−1) ((tan ϕ)/e))))]))))=(((sin^2 ϕ+e^2 cos^2 ϕ)[1+(1/(tan^2 (α−ϕ)))])/([1+(1/(tan^2 (α+tan^(−1) ((tan ϕ)/e))))])) ...(i) u^2 =v^2 +2g(s sin α−h) with U=((2gh)/u^2 ) ((1/U)+1)V=1+((2 tan α)/(1+(1/(tan^2 (α−ϕ))))){tan α+(1/(tan (α−ϕ)))+(√([tan α+(1/(tan (α−ϕ)))]^2 −V[1+(1/(tan^2 (α−ϕ)))]))}$
	$A B = s$ $v_{1} \sin φ = v_{2} \sin ϕ$ ${e v}_{1} \cos φ = v_{2} \cos ϕ$ $\frac{\tan φ}{e} = \tan ϕ \Rightarrow \sin ϕ = \frac{\tan φ}{\sqrt{e^{2} + \tan^{2} φ}} = \frac{1}{\sqrt{1 + \frac{e^{2}}{\tan^{2} φ}}}$ $\Rightarrow ϕ = \tan^{- 1} \frac{\tan φ}{e}$ $\Rightarrow v_{2} = \frac{\sin φ}{\sin ϕ} v = \sqrt{\sin^{2} φ + e^{2} \cos^{2} φ} v$ $t = \frac{s \cos α}{v \cos (\frac{π}{2} - α + φ)} = \frac{s \cos α}{v \sin (α - φ)}$ $- s \sin α + h = v \cos (α - φ) t - \frac{{g t}^{2}}{2}$ $\frac{{g s}^{2} \cos^{2} α}{2 v^{2}} [1 + \frac{1}{\tan^{2} (α - φ)}] - [\tan α + \frac{1}{\tan (α - φ)}] s \cos α + h = 0$ $\Rightarrow \frac{s \cos α}{2 h} = \frac{1}{\frac{2 g h}{v^{2}} [1 + \frac{1}{\tan^{2} (α - φ)}]} {\tan α + \frac{1}{\tan (α - φ)} + \sqrt{{[\tan α + \frac{1}{\tan (α - φ)}]}^{2} - \frac{2 g h}{v^{2}} [1 + \frac{1}{\tan^{2} (α - φ)}]}}$ $w i t h V = \frac{2 g h}{v^{2}}$ $\Rightarrow \frac{s \cos α}{2 h} = \frac{1}{V [1 + \frac{1}{\tan^{2} (α - φ)}]} {\tan α + \frac{1}{\tan (α - φ)} + \sqrt{{[\tan α + \frac{1}{\tan (α - φ)}]}^{2} - V [1 + \frac{1}{\tan^{2} (α - φ)}]}}$ $t = \frac{s \cos α}{v_{2} \cos (\frac{π}{2} - α - ϕ)} = \frac{s \cos α}{v \sin (α + ϕ) \sqrt{\sin^{2} φ + e^{2} \cos^{2} φ}}$ $- s \sin α + h = v_{2} \cos (α + ϕ) t - \frac{{g t}^{2}}{2}$ $- s \sin α + h = \frac{s \cos α}{\tan (α + ϕ)} - \frac{{g s}^{2} \cos^{2} α}{2 v^{2} (\sin^{2} φ + e^{2} \cos^{2} φ)} [1 + \frac{1}{\tan^{2} (α + ϕ)}]$ $\frac{{g s}^{2} \cos^{2} α}{2 v^{2} (\sin^{2} φ + e^{2} \cos^{2} φ)} [1 + \frac{1}{\tan^{2} (α + ϕ)}] - [\tan α + \frac{1}{\tan (α + ϕ)}] s \cos α + h = 0$ $\Rightarrow \frac{s \cos α}{2 h} = \frac{1}{\frac{2 g h}{v^{2} (\sin^{2} φ + e^{2} \cos^{2} φ)} [1 + \frac{1}{\tan^{2} (α + \tan^{- 1} \frac{\tan φ}{e})}]} {\tan α + \frac{1}{\tan (α + \tan^{- 1} \frac{\tan φ}{e})} + \sqrt{{[\tan α + \frac{1}{\tan (α + \tan^{- 1} \frac{\tan φ}{e})}]}^{2} - \frac{2 g h}{v^{2} (\sin^{2} φ + e^{2} \cos^{2} φ)} [1 + \frac{1}{\tan^{2} (α + \tan^{- 1} \frac{\tan φ}{e})}]}}$ $\Rightarrow \frac{s \cos α}{2 h} = \frac{1}{\frac{V}{(\sin^{2} φ + e^{2} \cos^{2} φ)} [1 + \frac{1}{\tan^{2} (α + \tan^{- 1} \frac{\tan φ}{e})}]} {\tan α + \frac{1}{\tan (α + \tan^{- 1} \frac{\tan φ}{e})} + \sqrt{{[\tan α + \frac{1}{\tan (α + \tan^{- 1} \frac{\tan φ}{e})}]}^{2} - \frac{V}{(\sin^{2} φ + e^{2} \cos^{2} φ)} [1 + \frac{1}{\tan^{2} (α + \tan^{- 1} \frac{\tan φ}{e})}]}}$ $\frac{1}{\frac{V}{(\sin^{2} φ + e^{2} \cos^{2} φ)} [1 + \frac{1}{\tan^{2} (α + \tan^{- 1} \frac{\tan φ}{e})}]} {\tan α + \frac{1}{\tan (α + \tan^{- 1} \frac{\tan φ}{e})} + \sqrt{{[\tan α + \frac{1}{\tan (α + \tan^{- 1} \frac{\tan φ}{e})}]}^{2} - \frac{V}{(\sin^{2} φ + e^{2} \cos^{2} φ)} [1 + \frac{1}{\tan^{2} (α + \tan^{- 1} \frac{\tan φ}{e})}]}} = \frac{1}{V [1 + \frac{1}{\tan^{2} (α - φ)}]} {\tan α + \frac{1}{\tan (α - φ)} + \sqrt{{[\tan α + \frac{1}{\tan (α - φ)}]}^{2} - V [1 + \frac{1}{\tan^{2} (α - φ)}]}}$ $\Rightarrow \frac{\tan α + \frac{1}{\tan (α - φ)} + \sqrt{{[\tan α + \frac{1}{\tan (α - φ)}]}^{2} - V [1 + \frac{1}{\tan^{2} (α - φ)}]}}{\tan α + \frac{1}{\tan (α + \tan^{- 1} \frac{\tan φ}{e})} + \sqrt{{[\tan α + \frac{1}{\tan (α + \tan^{- 1} \frac{\tan φ}{e})}]}^{2} - \frac{V}{(\sin^{2} φ + e^{2} \cos^{2} φ)} [1 + \frac{1}{\tan^{2} (α + \tan^{- 1} \frac{\tan φ}{e})}]}} = \frac{(\sin^{2} φ + e^{2} \cos^{2} φ) [1 + \frac{1}{\tan^{2} (α - φ)}]}{[1 + \frac{1}{\tan^{2} (α + \tan^{- 1} \frac{\tan φ}{e})}]} . . . (i)$ $u^{2} = v^{2} + 2 g (s \sin α - h)$ $w i t h U = \frac{2 g h}{u^{2}}$ $(\frac{1}{U} + 1) V = 1 + \frac{2 \tan α}{1 + \frac{1}{\tan^{2} (α - φ)}} {\tan α + \frac{1}{\tan (α - φ)} + \sqrt{{[\tan α + \frac{1}{\tan (α - φ)}]}^{2} - V [1 + \frac{1}{\tan^{2} (α - φ)}]}}$
	Commented by mr W last updated on 04/Nov/19

	$b y s o l v i n g t h i s q u a d r a t i c e q n . f o r V$ $w e g e t V = f (U) a n d p u t i t i n t o (i) t o$ $h a v e a n e q n . w i t h U a n d φ .$
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