Question Number 72563 by ajfour last updated on 30/Oct/19
Commented by ajfour last updated on 30/Oct/19
$${Find}\:{u}_{{min}} \:{for}\:{the}\:{stone}\:{to}\:{hit} \\ $$$${the}\:{sphere}\:{anywhere}. \\ $$
Answered by mr W last updated on 30/Oct/19
Commented by mr W last updated on 30/Oct/19
$${x}={u}\:\mathrm{cos}\:\theta\:{t} \\ $$$${y}={u}\:\mathrm{sin}\:\theta\:{t}−\frac{{gt}^{\mathrm{2}} }{\mathrm{2}} \\ $$$${y}={x}\:\mathrm{tan}\:\theta−\frac{{g}\left(\mathrm{1}+\mathrm{tan}^{\mathrm{2}} \:\theta\right){x}^{\mathrm{2}} }{\mathrm{2}{u}^{\mathrm{2}} } \\ $$$${with}\:{t}=\mathrm{tan}\:\theta \\ $$$$\Rightarrow{y}={tx}−\frac{{g}\left(\mathrm{1}+\mathrm{t}^{\mathrm{2}} \right){x}^{\mathrm{2}} }{\mathrm{2}{u}^{\mathrm{2}} } \\ $$$${y}'={t}−\frac{{g}\left(\mathrm{1}+{t}^{\mathrm{2}} \right){x}}{{u}^{\mathrm{2}} } \\ $$$${at}\:{point}\:{B}\left({d}−{R}\:\mathrm{sin}\:\varphi,\:{R}\left(\mathrm{1}−\mathrm{cos}\:\varphi\right)\right): \\ $$$${R}\left(\mathrm{1}−\mathrm{cos}\:\varphi\right)={t}\left({d}−{R}\:\mathrm{sin}\:\varphi\right)−\frac{{g}\left(\mathrm{1}+\mathrm{t}^{\mathrm{2}} \right)\left({d}−{R}\:\mathrm{sin}\:\varphi\right)^{\mathrm{2}} }{\mathrm{2}{u}^{\mathrm{2}} }\:\:\:…\left({i}\right) \\ $$$$−\mathrm{tan}\:\varphi={t}−\frac{{g}\left(\mathrm{1}+{t}^{\mathrm{2}} \right)\left({d}−{R}\:\mathrm{sin}\:\varphi\right)}{{u}^{\mathrm{2}} }\:\:\:…\left({ii}\right) \\ $$$$\Rightarrow\frac{{g}\left(\mathrm{1}+{t}^{\mathrm{2}} \right)\left({d}−{R}\:\mathrm{sin}\:\varphi\right)}{{u}^{\mathrm{2}} }={t}+\mathrm{tan}\:\varphi \\ $$$${R}\left(\mathrm{1}−\mathrm{cos}\:\varphi\right)={t}\left({d}−{R}\:\mathrm{sin}\:\varphi\right)−\frac{\left({d}−{R}\:\mathrm{sin}\:\varphi\right)\left({t}+\mathrm{tan}\:\varphi\right)}{\mathrm{2}} \\ $$$${with}\:\delta=\frac{{d}}{{R}} \\ $$$$\Rightarrow{t}=\mathrm{tan}\:\varphi+\frac{\mathrm{2}\left(\mathrm{1}−\mathrm{cos}\:\varphi\right)}{\delta−\mathrm{sin}\:\varphi} \\ $$$$\Rightarrow\frac{{gR}}{{u}^{\mathrm{2}} }=\frac{{t}+\mathrm{tan}\:\varphi}{\left(\delta−\mathrm{sin}\:\varphi\right)\left(\mathrm{1}+{t}^{\mathrm{2}} \right)} \\ $$$${let}\:{U}=\frac{{gR}}{\mathrm{2}{u}^{\mathrm{2}} } \\ $$$$\Rightarrow\mathrm{2}{U}=\frac{{t}+\mathrm{tan}\:\varphi}{\left(\delta−\mathrm{sin}\:\varphi\right)\left(\mathrm{1}+{t}^{\mathrm{2}} \right)} \\ $$$$\Rightarrow\mathrm{2}{U}=\frac{\mathrm{2}\:\mathrm{tan}\:\varphi+\frac{\mathrm{2}\left(\mathrm{1}−\mathrm{cos}\:\varphi\right)}{\delta−\mathrm{sin}\:\varphi}}{\left(\delta−\mathrm{sin}\:\varphi\right)\left[\mathrm{1}+\left(\mathrm{tan}\:\varphi+\frac{\mathrm{2}\left(\mathrm{1}−\mathrm{cos}\:\varphi\right)}{\delta−\mathrm{sin}\:\varphi}\right)^{\mathrm{2}} \right]} \\ $$$$\Rightarrow{U}=\frac{\mathrm{1}−\mathrm{cos}\:\varphi+\left(\delta−\mathrm{sin}\:\varphi\right)\:\mathrm{tan}\:\varphi}{\left(\delta−\mathrm{sin}\:\varphi\right)^{\mathrm{2}} +\left[\left(\delta−\mathrm{sin}\:\varphi\right)\:\mathrm{tan}\:\varphi+\mathrm{2}\left(\mathrm{1}−\mathrm{cos}\:\varphi\right)\right]^{\mathrm{2}} } \\ $$$$\frac{{dU}}{{d}\varphi}=\mathrm{0}\:\Rightarrow\:\varphi\:\Rightarrow\:{U}_{{max}} \Rightarrow\:{u}_{{min}} \\ $$$$ \\ $$$${example}: \\ $$$$\delta=\frac{{d}}{{R}}=\frac{\mathrm{6}}{\mathrm{2}}=\mathrm{3} \\ $$$$\Rightarrow\varphi=\mathrm{0}.\mathrm{7309} \\ $$$$\Rightarrow{U}_{{max}} =\mathrm{0}.\mathrm{1922}\:\Rightarrow{u}_{{min}} =\mathrm{1}.\mathrm{613}\sqrt{{gR}} \\ $$
Commented by mr W last updated on 30/Oct/19
Commented by ajfour last updated on 30/Oct/19
$${Thanks}\:{Sir},\:{but}\:{lets}\:{see}\:{if}\:{a} \\ $$$${final}\:{answer}\:{be}\:{possible}.. \\ $$
Commented by malwaan last updated on 30/Oct/19
$$\boldsymbol{{I}}\:\boldsymbol{{think}} \\ $$$$\boldsymbol{{u}}_{\boldsymbol{{max}}} \:\boldsymbol{{at}}\:\boldsymbol{\theta}=\:\mathrm{45}^{°} \\ $$
Commented by mr W last updated on 30/Oct/19
$${it}\:{is}\:{to}\:{find}\:{a}\:{parabola}\:{which}\:{tangents} \\ $$$${the}\:{circle}\:{AND}\:{has}\:{the}\:{minimum} \\ $$$${energy}.\:\theta\:{is}\:{not}\:{a}\:{constant},\:{but} \\ $$$${depends}\:{on}\:{the}\:{position}\:{of}\:{the}\:{circle}. \\ $$
Commented by mr W last updated on 30/Oct/19
Commented by mr W last updated on 30/Oct/19