Question Number 15932 by Tinkutara last updated on 15/Jun/17 | ||
$$\mathrm{A}\:\mathrm{particle}\:\mathrm{is}\:\mathrm{projected}\:\mathrm{from}\:\mathrm{the}\:\mathrm{foot}\:\mathrm{of} \\ $$ $$\mathrm{an}\:\mathrm{inclined}\:\mathrm{plane}\:\mathrm{having}\:\mathrm{inclination} \\ $$ $$\mathrm{45}°,\:\mathrm{with}\:\mathrm{the}\:\mathrm{velocity}\:{u}\:\mathrm{at}\:\mathrm{an}\:\mathrm{angle}\:\theta \\ $$ $$\left(>\:\mathrm{45}°\right)\:\mathrm{with}\:\mathrm{the}\:\mathrm{horizontal}\:\mathrm{in}\:\mathrm{a}\:\mathrm{vertical} \\ $$ $$\mathrm{plane}\:\mathrm{containing}\:\mathrm{the}\:\mathrm{line}\:\mathrm{of}\:\mathrm{greatest} \\ $$ $$\mathrm{slope}\:\mathrm{through}\:\mathrm{the}\:\mathrm{point}\:\mathrm{of}\:\mathrm{projection}. \\ $$ $$\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{tan}\:\theta\:\mathrm{if}\:\mathrm{the}\:\mathrm{particle} \\ $$ $$\mathrm{strikes}\:\mathrm{the}\:\mathrm{plane} \\ $$ $$\left(\mathrm{i}\right)\:\mathrm{Horizontally} \\ $$ $$\left(\mathrm{ii}\right)\:\mathrm{Normally} \\ $$ | ||
Commented byTinkutara last updated on 15/Jun/17 | ||
Answered by ajfour last updated on 15/Jun/17 | ||
$${when}\:{it}\:{hits}\:{the}\:{incline}\:{horizontally} \\ $$ $${it}\:{is}\:{at}\:{its}\:{max}.\:{hright}. \\ $$ $$\mathrm{tan}\:\alpha=\frac{{H}}{\left({R}/\mathrm{2}\right)}\:\:\:\:,\:{and}\:{witb}\:\alpha=\mathrm{45}° \\ $$ $$\:{R}\:=\:\mathrm{2}{H} \\ $$ $$\:\frac{\mathrm{2}{u}^{\mathrm{2}} \mathrm{sin}\:\theta\mathrm{cos}\:\theta}{{g}}=\frac{\mathrm{2}{u}^{\mathrm{2}} \mathrm{sin}\:^{\mathrm{2}} \theta}{\mathrm{2}{g}} \\ $$ $$\Rightarrow\:\:\mathrm{tan}\:\theta\:=\:\mathrm{2}\:\:{or}\:\:\theta=\mathrm{tan}^{−\mathrm{1}} \left(\mathrm{2}\right)\:. \\ $$ $${when}\:{it}\:{strikes}\:{nirmally}, \\ $$ $${its}\:{component}\:{of}\:{velocity}\: \\ $$ $${parallel}\:{to}\:{incline}\:{becomes}\:{zero} \\ $$ $${by}\:{the}\:{time}\:{its}\:{displacement} \\ $$ $${perpendicular}\:{to}\:{incline}\:{becomes} \\ $$ $${zero}. \\ $$ $$\:{v}_{{x}'} ={u}\mathrm{cos}\:\left(\theta−\alpha\right)−{gt}\mathrm{sin}\:\alpha=\mathrm{0} \\ $$ $${or}\:\:{t}=\frac{{u}\mathrm{cos}\:\left(\theta−\alpha\right)}{{g}\mathrm{sin}\:\alpha}\:\:\:\:\:\:.....\left({i}\right) \\ $$ $$\:{s}_{{y}'} ={ut}\mathrm{sin}\:\left(\theta−\alpha\right)−\frac{\mathrm{1}}{\mathrm{2}}{gt}^{\mathrm{2}} \mathrm{cos}\:\alpha=\mathrm{0} \\ $$ $$\Rightarrow\:{t}=\frac{\mathrm{2}{u}\mathrm{sin}\:\left(\theta−\alpha\right)}{{g}\mathrm{cos}\:\alpha}\:\:\:\:\:\:\:\:...\left({ii}\right) \\ $$ $${equating}\:\left({ii}\right)\:{and}\:\left({i}\right): \\ $$ $$\:\frac{\mathrm{2}{u}\mathrm{sin}\:\left(\theta−\alpha\right)}{{g}\mathrm{cos}\:\alpha}=\frac{{u}\mathrm{cos}\:\left(\theta−\alpha\right)}{{g}\mathrm{sin}\:\alpha} \\ $$ $$\Rightarrow\:\:\:\mathrm{tan}\:\left(\theta−\alpha\right)\mathrm{tan}\:\alpha\:=\frac{\mathrm{1}}{\mathrm{2}} \\ $$ $$\:\:{as}\:\alpha=\mathrm{45}°,\:\:\:\mathrm{tan}\:\left(\theta−\mathrm{45}°\right)×\mathrm{1}=\frac{\mathrm{1}}{\mathrm{2}} \\ $$ $$\:\theta=\:\mathrm{45}°+\mathrm{tan}^{−\mathrm{1}} \left(\frac{\mathrm{1}}{\mathrm{2}}\right)\:\:. \\ $$ | ||
Commented byTinkutara last updated on 16/Jun/17 | ||
$$\mathrm{Thanks}\:\mathrm{Sir}! \\ $$ | ||