Question Number 13705 by ajfour last updated on 22/May/17
$${Assuming}\:{no}\:{air}\:{resistance}, \\ $$$${and}\:{angle}\:{of}\:{projection}\:\alpha=\frac{\pi}{\mathrm{4}}\:, \\ $$$${find}\:{the}\:{ratio}\:{of}\:{the}\:{length}\:{of} \\ $$$${trajectory}\:\boldsymbol{{L}}\:{of}\:{a}\:{projectile}\:{motion}\: \\ $$$$\left({by}\:{the}\:{time}\:{it}\:{hits}\:{the}\:{ground}\right) \\ $$$${to}\:{its}\:{horizontal}\:{range}\:\boldsymbol{{R}}\:{on}\: \\ $$$${ground}.\:\:\:\:\:\:\frac{\boldsymbol{{L}}}{\boldsymbol{{R}}}=? \\ $$
Commented by ajfour last updated on 22/May/17
Commented by mrW1 last updated on 22/May/17
$${for}\:{parabola}\:{f}\left({x}\right)={ax}^{\mathrm{2}} \\ $$$${the}\:{length}\:{of}\:{the}\:{curve}\:{from}\:{x}=\mathrm{0}\:{to} \\ $$$${x}={b}\:{is} \\ $$$${l}=\int_{\mathrm{0}} ^{{b}} \sqrt{\mathrm{1}+\left(\frac{{dy}}{{dx}}\right)^{\mathrm{2}} }{dx} \\ $$$$=\int_{\mathrm{0}} ^{{b}} \sqrt{\mathrm{1}+\left(\mathrm{2}{ax}\right)^{\mathrm{2}} }{dx} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}{a}}\int_{\mathrm{0}} ^{\mathrm{2}{ab}} \sqrt{\mathrm{1}+{t}^{\mathrm{2}} }{dt} \\ $$$$=\frac{\mathrm{1}}{\mathrm{4}{a}}\left[{t}\sqrt{\mathrm{1}+{t}^{\mathrm{2}} }+\mathrm{ln}\:\left({t}+\sqrt{\mathrm{1}+{t}^{\mathrm{2}} }\right)\right]_{\mathrm{0}} ^{\mathrm{2}{ab}} \\ $$$$=\frac{{b}}{\mathrm{2}}×\frac{\mathrm{1}}{\mathrm{2}{ab}}\left(\mathrm{2}{ab}\sqrt{\mathrm{1}+\left(\mathrm{2}{ab}\right)^{\mathrm{2}} }+\mathrm{ln}\:\sqrt{\mathrm{1}+\left(\mathrm{2}{ab}\right)^{\mathrm{2}} }\right) \\ $$$$ \\ $$$${apply}\:{this}\:{in}\:{our}\:{case}: \\ $$$${b}=\frac{{R}}{\mathrm{2}} \\ $$$${height}\:{of}\:{parabola}\:{h}\:{is} \\ $$$${h}=\frac{{R}}{\mathrm{4}}×\mathrm{tan}\:\alpha_{\mathrm{0}} =\frac{{R}}{\mathrm{4}}×\mathrm{tan}\:\frac{\pi}{\mathrm{4}}=\frac{{R}}{\mathrm{4}} \\ $$$${h}={a}\left(\frac{{R}}{\mathrm{2}}\right)^{\mathrm{2}} ={a}\frac{{R}^{\mathrm{2}} }{\mathrm{4}}=\frac{{R}}{\mathrm{4}} \\ $$$$\Rightarrow{a}=\frac{\mathrm{1}}{{R}} \\ $$$$\mathrm{2}{ab}=\mathrm{2}×\frac{\mathrm{1}}{{R}}×\frac{{R}}{\mathrm{2}}=\mathrm{1} \\ $$$$ \\ $$$$\Rightarrow{L}=\mathrm{2}×\frac{{R}}{\mathrm{4}}×\frac{\mathrm{1}}{\mathrm{1}}×\left[\mathrm{1}×\sqrt{\mathrm{1}+\mathrm{1}^{\mathrm{2}} }+\mathrm{ln}\:\left(\mathrm{1}+\sqrt{\mathrm{1}+\mathrm{1}^{\mathrm{2}} }\right)\right]=\frac{{R}}{\mathrm{2}}×\left[\sqrt{\mathrm{2}}+\mathrm{ln}\:\left(\mathrm{1}+\sqrt{\mathrm{2}}\right)\right] \\ $$$$\Rightarrow\frac{{L}}{{R}}=\frac{\sqrt{\mathrm{2}}+\mathrm{ln}\:\left(\mathrm{1}+\sqrt{\mathrm{2}}\right)}{\mathrm{2}}\approx\mathrm{1}.\mathrm{15} \\ $$
Commented by ajfour last updated on 22/May/17
$${thank}\:{you}\:{Sir},\:{our}\:{answers}\:{even} \\ $$$${match}! \\ $$
Answered by ajfour last updated on 22/May/17
$${dL}=\sqrt{\mathrm{1}+\left(\frac{{dy}}{{dx}}\right)^{\mathrm{2}} }\:{dx} \\ $$$${dy}=\left({u}\mathrm{sin}\:\alpha−{gt}\right){dt} \\ $$$${dx}=\left({u}\mathrm{cos}\:\alpha\right){dt} \\ $$$$\frac{{dy}}{{dx}}=\left(\frac{{u}\mathrm{sin}\:\alpha−{gt}}{{u}\mathrm{cos}\:\alpha}\right) \\ $$$$\int_{\mathrm{0}} ^{\:\:\boldsymbol{{L}}} {dL}=\int_{\mathrm{0}} ^{\:\:\boldsymbol{{T}}} \sqrt{\mathrm{1}+\left(\frac{{u}\mathrm{sin}\:\alpha−{gt}}{{u}\mathrm{cos}\:\alpha}\right)^{\mathrm{2}} }\left({u}\mathrm{cos}\:\alpha\right){dt} \\ $$$${for}\:{y}\:{component}\:{of}\:{motion} \\ $$$$\boldsymbol{{v}}_{\boldsymbol{{y}}} =\boldsymbol{{u}}\mathrm{sin}\:\boldsymbol{\alpha}−{gt} \\ $$$$\:\:\:\:\:=\mathrm{0}\:\:\:{for}\:\:{t}=\frac{{T}}{\mathrm{2}} \\ $$$$\Rightarrow\:\boldsymbol{{T}}=\frac{\mathrm{2}\boldsymbol{{u}}\mathrm{sin}\:\boldsymbol{\alpha}}{\boldsymbol{{g}}} \\ $$$${for}\:\:\boldsymbol{\alpha}=\frac{\boldsymbol{\pi}}{\mathrm{4}}\:\:,\:\:{T}=\frac{\boldsymbol{{u}}\sqrt{\mathrm{2}}}{\boldsymbol{{g}}}\:\:\:\:…..\left(\boldsymbol{{i}}\right) \\ $$$$\boldsymbol{{L}}=\frac{{u}}{\:\sqrt{\mathrm{2}}}\int_{\mathrm{0}} ^{\:\:{u}\sqrt{\mathrm{2}}/{g}} \sqrt{\mathrm{1}+\left(\mathrm{1}−\frac{{gt}\sqrt{\mathrm{2}}}{{u}}\right)^{\mathrm{2}} }{dt} \\ $$$$\:\:\:{let}\:\:\boldsymbol{{p}}=\mathrm{1}−\frac{\boldsymbol{{gt}}\sqrt{\mathrm{2}}}{\boldsymbol{{u}}} \\ $$$$\:\:\boldsymbol{{dp}}=−\frac{\boldsymbol{{g}}\sqrt{\mathrm{2}}}{\boldsymbol{{u}}}\boldsymbol{{dt}}\:\:\:\Rightarrow\:{dt}=−\frac{\boldsymbol{{u}}}{\boldsymbol{{g}}\sqrt{\mathrm{2}}}\boldsymbol{{dp}} \\ $$$${when}\:{t}=\mathrm{0},\:\:\boldsymbol{{p}}=\mathrm{1} \\ $$$${when}\:{t}=\frac{{u}\sqrt{\mathrm{2}}}{{g}}\:,\:\:\boldsymbol{{p}}=−\mathrm{1} \\ $$$$\therefore\:\boldsymbol{{L}}=\frac{{u}}{\:\sqrt{\mathrm{2}}}\int_{\mathrm{1}} ^{\:\:−\mathrm{1}} \sqrt{\mathrm{1}+\boldsymbol{{p}}^{\mathrm{2}} }\left(−\frac{\boldsymbol{{u}}}{\boldsymbol{{g}}\sqrt{\mathrm{2}}}\right)\boldsymbol{{dp}} \\ $$$$\boldsymbol{{L}}=\frac{\boldsymbol{{u}}^{\mathrm{2}} }{\boldsymbol{{g}}}\int_{\mathrm{0}} ^{\:\:\mathrm{1}} \sqrt{\mathrm{1}+\boldsymbol{{p}}^{\mathrm{2}} }\boldsymbol{{dp}} \\ $$$$\boldsymbol{{L}}=\frac{\boldsymbol{{u}}^{\mathrm{2}} }{\boldsymbol{{g}}}\left[\frac{\boldsymbol{{p}}}{\mathrm{2}}\sqrt{\mathrm{1}+\boldsymbol{{p}}^{\mathrm{2}} }+\frac{\mathrm{1}}{\mathrm{2}}\mathrm{ln}\:\left(\boldsymbol{{p}}+\sqrt{\mathrm{1}+\boldsymbol{{p}}^{\mathrm{2}} }\right)\right]_{\mathrm{0}} ^{\mathrm{1}} \\ $$$$\boldsymbol{{L}}=\frac{\boldsymbol{{u}}^{\mathrm{2}} }{\mathrm{2}\boldsymbol{{g}}}\left[\sqrt{\mathrm{2}}+\mathrm{ln}\:\left(\mathrm{1}+\sqrt{\mathrm{2}}\right)\right] \\ $$$$\boldsymbol{{R}}=\left(\boldsymbol{{u}}\mathrm{cos}\:\boldsymbol{\alpha}\right)\boldsymbol{{T}} \\ $$$$\:\:\:\:=\frac{\boldsymbol{{u}}}{\:\sqrt{\mathrm{2}}}\:\frac{\boldsymbol{{u}}\sqrt{\mathrm{2}}}{\boldsymbol{{g}}}=\frac{\boldsymbol{{u}}^{\mathrm{2}} }{\boldsymbol{{g}}} \\ $$$$\boldsymbol{{so}},\:\:\frac{\boldsymbol{{L}}}{\boldsymbol{{R}}}=\frac{\sqrt{\mathrm{2}}+\mathrm{ln}\:\left(\mathrm{1}+\sqrt{\mathrm{2}}\right)}{\mathrm{2}}\:\approx\mathrm{1}.\mathrm{148} \\ $$