Question Number 46482 by peter frank last updated on 27/Oct/18
Commented by peter frank last updated on 27/Oct/18
$$\mathrm{help} \\ $$$$ \\ $$
Answered by tanmay.chaudhury50@gmail.com last updated on 28/Oct/18
$${horizontal}\:{velocity}\:{of}\:{projection}={vcos}\alpha \\ $$$${vertical}\:{velocity}\:{of}\:{projection}={vsin}\alpha \\ $$$${eqn}\:{for}\:{horigontal}\:{motion} \\ $$$${x}=\left({vcos}\alpha\right){t} \\ $$$${eqn}\:{for}\:{vdrtical}\:{motion} \\ $$$${y}=\left({vsin}\alpha\right){t}−\frac{\mathrm{1}}{\mathrm{2}}{gt}^{\mathrm{2}} \\ $$$${there}\:{is}\:{no}\:{accelartion}\:{along}\:{horizontal}\:{direction} \\ $$$${accelaration}\:\left({g}\right)\:{is}\:{acting}\:{vertically}\:{down} \\ $$$${after}\:{time}\:{t}_{\mathrm{0}} \:\:{the}\:{vertical}\:{velocity}=\mathrm{0} \\ $$$$\mathrm{0}={vsin}\alpha−{gt}_{\mathrm{0}} \:\:\:\:{t}_{\mathrm{0}} =\frac{{vsin}\alpha}{{g}} \\ $$$${total}\:{time}\:{of}\:{flight}=\mathrm{2}{t}_{\mathrm{0}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\frac{\mathrm{2}{vsin}\alpha}{{g}} \\ $$$${Range}=\left({vcos}\alpha\right)\left(\frac{\mathrm{2}{vsin}\alpha}{{g}}\right)={v}^{\mathrm{2}} ×\frac{\mathrm{2}{sin}\alpha{cos}\alpha}{{g}}=\frac{{v}^{\mathrm{2}} {sin}\mathrm{2}\alpha}{{g}} \\ $$$${greatest}\:{height}\:{when}\:{vertical}\:{component}\:{of} \\ $$$${velocity}\:{is}\:{zero}. \\ $$$$\mathrm{0}^{\mathrm{2}} =\left({vsin}\alpha\right)^{\mathrm{2}} −\mathrm{2}{g}\boldsymbol{{H}}\:\:\left[{formula}\:{v}^{\mathrm{2}} ={u}^{\mathrm{2}} −\mathrm{2}{as}\right] \\ $$$$\boldsymbol{{H}}=\frac{{v}^{\mathrm{2}} {sin}^{\mathrm{2}} \alpha}{\mathrm{2}{g}} \\ $$$${y}=\left({vsin}\alpha\right){t}−\frac{\mathrm{1}}{\mathrm{2}}{gt}^{\mathrm{2}} \\ $$$${y}=\left({vsin}\alpha\right)\left(\frac{{x}}{{vcos}\alpha}\right)−\frac{\mathrm{1}}{\mathrm{2}}{g}\left(\frac{{x}}{{vcos}\alpha}\right)^{\mathrm{2}} \\ $$$${y}={xtan}\alpha−\frac{\mathrm{1}}{\mathrm{2}}{g}\left(\frac{{x}^{\mathrm{2}} }{{v}^{\mathrm{2}} {cos}^{\mathrm{2}} \alpha}\right) \\ $$$$ \\ $$$$ \\ $$$${y}={Ax}−{Bx}^{\mathrm{2}} \\ $$$${Bx}^{\mathrm{2}} −{Ax}=−{y} \\ $$$${x}^{\mathrm{2}} −\frac{{A}}{{B}}{x}=−\frac{{y}}{{B}} \\ $$$${x}^{\mathrm{2}} −\mathrm{2}.{x}.\frac{{A}}{\mathrm{2}{B}}+\frac{{A}^{\mathrm{2}} }{\mathrm{4}{B}^{\mathrm{2}} }=−\frac{{y}}{{B}}+\frac{{A}^{\mathrm{2}} }{\mathrm{4}{B}^{\mathrm{2}} } \\ $$$$\left({x}−\frac{{A}}{\mathrm{2}{B}}\right)^{\mathrm{2}} =\left(\frac{{A}^{\mathrm{2}} −\mathrm{4}{yB}}{\mathrm{4}{B}^{\mathrm{2}} }\right) \\ $$$$\left({x}−\frac{{A}}{\mathrm{2}{B}}\right)^{\mathrm{2}} =\left(−\mathrm{4}{B}\right)×\frac{\mathrm{1}}{\mathrm{4}{B}^{\mathrm{2}} }\left({y}−\frac{{A}^{\mathrm{2}} }{\mathrm{4}{B}}\right) \\ $$$$\left({x}−\frac{{A}}{\mathrm{2}{B}}\right)^{\mathrm{2}} =\left(−\mathrm{4}\right)×\left(\frac{\mathrm{1}}{\mathrm{4}{B}}\right)\left({y}−\frac{{A}^{\mathrm{2}} }{\mathrm{4}{B}}\right) \\ $$$${X}^{\mathrm{2}} =−\mathrm{4}{PY} \\ $$$${focus}\left(\mathrm{0},−{P}\right) \\ $$$${x}−\frac{{A}}{\mathrm{2}{B}}=\mathrm{0}\:\:\:\:{x}=\frac{{A}}{\mathrm{2}{B}}\:\:\: \\ $$$${y}−\frac{{A}^{\mathrm{2}} }{\mathrm{4}{B}}=−{P} \\ $$$${y}−\frac{{A}^{\mathrm{2}} }{\mathrm{4}{B}}=−\frac{\mathrm{1}}{\mathrm{4}{B}} \\ $$$${y}=\frac{{A}^{\mathrm{2}} −\mathrm{1}}{\mathrm{4}{B}} \\ $$$${focus}\left(\frac{{A}}{\mathrm{2}{B}},\frac{{A}^{\mathrm{2}} −\mathrm{1}}{\mathrm{4}{B}}\right) \\ $$$${directrix}\:\:{Y}={P} \\ $$$${y}−\frac{{A}^{\mathrm{2}} }{\mathrm{4}{B}}=\frac{\mathrm{1}}{\mathrm{4}{B}} \\ $$$${y}=\frac{{A}^{\mathrm{2}} +\mathrm{1}}{\mathrm{4}{B}} \\ $$$$ \\ $$$$ \\ $$
Commented by peter frank last updated on 28/Oct/18
$$\mathrm{how}\:\mathrm{do}\:\mathrm{i}\:\mathrm{get}\:\mathrm{focus}\:\mathrm{and}\:\mathrm{directrix} \\ $$
Commented by peter frank last updated on 28/Oct/18
$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you} \\ $$