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Question-23920




Question Number 23920 by ajfour last updated on 09/Nov/17
Commented by ajfour last updated on 09/Nov/17
Find u, 𝛉 for a projecile in terms  of a, b, and R such that it  goes from A to B through M ,  the midpoint of upper edge.  Assume the incline frictionless.
$${Find}\:\boldsymbol{{u}},\:\boldsymbol{\theta}\:{for}\:{a}\:{projecile}\:{in}\:{terms} \\ $$$${of}\:\boldsymbol{{a}},\:\boldsymbol{{b}},\:{and}\:\boldsymbol{{R}}\:{such}\:{that}\:{it} \\ $$$${goes}\:{from}\:{A}\:{to}\:{B}\:{through}\:{M}\:, \\ $$$${the}\:{midpoint}\:{of}\:{upper}\:{edge}. \\ $$$${Assume}\:{the}\:{incline}\:{frictionless}. \\ $$
Answered by Physics lover last updated on 10/Nov/17
x direction is along AB  and y along (√(a^2 +b^2 ))    ((height)/(range)) = ((tan θ)/4) ⇒ tan θ = ((4(√(a^2 +b^2 )))/R)    H = (u_y ^2 /(2∙ ∣a_y ∣)) ⇒ ((u^2  Sin^2  θ)/(2g((b/( (√(a^2 +b^2 ))))))) = (√(a^2 +b^2 ))  ⇒ u = (√((2gb)/({Sin^2  θ}))) = (√((2gb)/({((16(a^2  + b^2 ))/(16(a^2 +b^2 ) +R^2 ))})))  ⇒ u =(√((2gb[16(a^2 +b^2 )+R^2 ])/(16(a^2 +b^2 ))))
$${x}\:{direction}\:{is}\:{along}\:{AB} \\ $$$${and}\:{y}\:{along}\:\sqrt{{a}^{\mathrm{2}} +{b}^{\mathrm{2}} } \\ $$$$ \\ $$$$\frac{{height}}{{range}}\:=\:\frac{{tan}\:\theta}{\mathrm{4}}\:\Rightarrow\:{tan}\:\theta\:=\:\frac{\mathrm{4}\sqrt{{a}^{\mathrm{2}} +{b}^{\mathrm{2}} }}{{R}} \\ $$$$ \\ $$$${H}\:=\:\frac{{u}_{{y}} ^{\mathrm{2}} }{\mathrm{2}\centerdot\:\mid{a}_{{y}} \mid}\:\Rightarrow\:\frac{{u}^{\mathrm{2}} \:{Sin}\:^{\mathrm{2}} \:\theta}{\mathrm{2}{g}\left(\frac{{b}}{\:\sqrt{{a}^{\mathrm{2}} +{b}^{\mathrm{2}} }}\right)}\:=\:\sqrt{{a}^{\mathrm{2}} +{b}^{\mathrm{2}} } \\ $$$$\Rightarrow\:{u}\:=\:\sqrt{\frac{\mathrm{2}{gb}}{\left\{{Sin}^{\mathrm{2}} \:\theta\right\}}}\:=\:\sqrt{\frac{\mathrm{2}{gb}}{\left\{\frac{\mathrm{16}\left({a}^{\mathrm{2}} \:+\:{b}^{\mathrm{2}} \right)}{\mathrm{16}\left({a}^{\mathrm{2}} +{b}^{\mathrm{2}} \right)\:+{R}^{\mathrm{2}} }\right\}}} \\ $$$$\Rightarrow\:{u}\:=\sqrt{\frac{\mathrm{2}{gb}\left[\mathrm{16}\left({a}^{\mathrm{2}} +{b}^{\mathrm{2}} \right)+{R}^{\mathrm{2}} \right]}{\mathrm{16}\left({a}^{\mathrm{2}} +{b}^{\mathrm{2}} \right)}} \\ $$
Commented by ajfour last updated on 10/Nov/17
Very Nice. Thank you.
$${Very}\:{Nice}.\:{Thank}\:{you}. \\ $$
Commented by Physics lover last updated on 10/Nov/17
you are welcome,sir.
$${you}\:{are}\:{welcome},{sir}. \\ $$

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