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Question-25838




Question Number 25838 by ajfour last updated on 15/Dec/17
Commented by ajfour last updated on 15/Dec/17
(i) Determine ∠ θ such that   projectile hits inclined plane  perpendicularly at P.  (ii) Find the coefficient of  restitution e such that after   another bounce at Q projectile  returns to point of projection O.
$$\left({i}\right)\:{Determine}\:\angle\:\theta\:{such}\:{that}\: \\ $$$${projectile}\:{hits}\:{inclined}\:{plane} \\ $$$${perpendicularly}\:{at}\:{P}. \\ $$$$\left({ii}\right)\:{Find}\:{the}\:{coefficient}\:{of} \\ $$$${restitution}\:\boldsymbol{{e}}\:{such}\:{that}\:{after}\: \\ $$$${another}\:{bounce}\:{at}\:{Q}\:{projectile} \\ $$$${returns}\:{to}\:{point}\:{of}\:{projection}\:{O}. \\ $$
Commented by ajfour last updated on 16/Dec/17
not complete i suppose ?!
$${not}\:{complete}\:{i}\:{suppose}\:?! \\ $$
Commented by ajfour last updated on 17/Dec/17
(i)   To hit at P perpendicularly    ucos θ−(gsin α)T_0 = 0  , where    (usin θ)T_0  − (((gcos α)T_0 ^(  2) )/2) =0  so    tan θ = (1/(2tan α))  ⇒   θ=tan^(−1) ((1/(2tan α))) .
$$\left({i}\right)\:\:\:{To}\:{hit}\:{at}\:{P}\:{perpendicularly} \\ $$$$\:\:{u}\mathrm{cos}\:\theta−\left({g}\mathrm{sin}\:\alpha\right){T}_{\mathrm{0}} =\:\mathrm{0}\:\:,\:{where} \\ $$$$\:\:\left({u}\mathrm{sin}\:\theta\right){T}_{\mathrm{0}} \:−\:\frac{\left({g}\mathrm{cos}\:\alpha\right){T}_{\mathrm{0}} ^{\:\:\mathrm{2}} }{\mathrm{2}}\:=\mathrm{0} \\ $$$${so}\:\:\:\:\mathrm{tan}\:\theta\:=\:\frac{\mathrm{1}}{\mathrm{2tan}\:\alpha} \\ $$$$\Rightarrow\:\:\:\theta=\mathrm{tan}^{−\mathrm{1}} \left(\frac{\mathrm{1}}{\mathrm{2tan}\:\alpha}\right)\:. \\ $$
Commented by jota@ last updated on 17/Dec/17
Correct
$${Correct} \\ $$

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