Question Number 214093 by ajfour last updated on 27/Nov/24
Commented by ajfour last updated on 27/Nov/24
$${If}\:{A},\:{B},\:{C}\:{in}\:{some}\:{time}\:{come}\:{in}\:{eql}. \\ $$$$\bigtriangleup\:{position}\:{then}\:{how}\:{are}\:{v}\:{and}\:{u} \\ $$$${related}?\:{After}\:{what}\:{time}\:{this} \\ $$$${happens}.\:{At}\:{t}=\mathrm{0},\:{all}\:{are}\:\:{at}\:{the}\:{origin}. \\ $$
Answered by ajfour last updated on 27/Nov/24
![y_A =ut , x_B =vt x_C =Rsin{(((u+v)t)/R)} y_C =R−Rcos {(((u+v)t)/R)} M(((ut)/2), ((vt)/2)) tan φ=(v/u) line MC y−((ut)/2)=(v/u)(x−((vt)/2)) R−Rcos {(((u+v)t)/R)} =((ut)/2)+(v/u)[Rsin{(((u+v)t)/R)}−((vt)/2)] say ω=((u+v)/R) R(1−cos ωt)−Rsin ωt=(((u^2 −v^2 )/(2u)))t then we also must have {Rsin (ωt)−((vt)/2)}^2 +{R−Rcos (ωt)−((ut)/2)}^2 =(((3t^2 )/4))(u^2 +v^2 ) ⇒ {Rsin (ωt)−((vt)/2)}(√(1+(v^2 /u^2 )))=(((√3)/2)t)(√(u^2 +v^2 )) Rsin (ωt)−((vt)/2)=(((√3)ut)/2) ....](https://www.tinkutara.com/question/Q214095.png)
$${y}_{{A}} ={ut}\:\:,\:\:{x}_{{B}} ={vt} \\ $$$${x}_{{C}} ={R}\mathrm{sin}\left\{\frac{\left({u}+{v}\right){t}}{{R}}\right\} \\ $$$${y}_{{C}} ={R}−{R}\mathrm{cos}\:\left\{\frac{\left({u}+{v}\right){t}}{{R}}\right\} \\ $$$${M}\left(\frac{{ut}}{\mathrm{2}},\:\frac{{vt}}{\mathrm{2}}\right) \\ $$$$\mathrm{tan}\:\phi=\frac{{v}}{{u}} \\ $$$${line}\:{MC} \\ $$$${y}−\frac{{ut}}{\mathrm{2}}=\frac{{v}}{{u}}\left({x}−\frac{{vt}}{\mathrm{2}}\right) \\ $$$${R}−{R}\mathrm{cos}\:\left\{\frac{\left({u}+{v}\right){t}}{{R}}\right\} \\ $$$$\:\:\:\:\:\:=\frac{{ut}}{\mathrm{2}}+\frac{{v}}{{u}}\left[{R}\mathrm{sin}\left\{\frac{\left({u}+{v}\right){t}}{{R}}\right\}−\frac{{vt}}{\mathrm{2}}\right] \\ $$$${say}\:\omega=\frac{{u}+{v}}{{R}}\: \\ $$$${R}\left(\mathrm{1}−\mathrm{cos}\:\omega{t}\right)−{R}\mathrm{sin}\:\omega{t}=\left(\frac{{u}^{\mathrm{2}} −{v}^{\mathrm{2}} }{\mathrm{2}{u}}\right){t} \\ $$$$\:\:{then}\:{we}\:{also}\:{must}\:{have} \\ $$$$\left\{{R}\mathrm{sin}\:\left(\omega{t}\right)−\frac{{vt}}{\mathrm{2}}\right\}^{\mathrm{2}} +\left\{{R}−{R}\mathrm{cos}\:\left(\omega{t}\right)−\frac{{ut}}{\mathrm{2}}\right\}^{\mathrm{2}} \\ $$$$\:\:\:=\left(\frac{\mathrm{3}{t}^{\mathrm{2}} }{\mathrm{4}}\right)\left({u}^{\mathrm{2}} +{v}^{\mathrm{2}} \right) \\ $$$$\Rightarrow\:\:\left\{{R}\mathrm{sin}\:\left(\omega{t}\right)−\frac{{vt}}{\mathrm{2}}\right\}\sqrt{\mathrm{1}+\frac{{v}^{\mathrm{2}} }{{u}^{\mathrm{2}} }}=\left(\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}{t}\right)\sqrt{{u}^{\mathrm{2}} +{v}^{\mathrm{2}} } \\ $$$$\:{R}\mathrm{sin}\:\left(\omega{t}\right)−\frac{{vt}}{\mathrm{2}}=\frac{\sqrt{\mathrm{3}}{ut}}{\mathrm{2}} \\ $$$$…. \\ $$