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Question Number 20915 by Tinkutara last updated on 07/Sep/17

Two shells are fired from a canon with speed u each, at angles of α and β respectively with the horizontal. The time interval between the shots is t. They collide in mid air after time T from the first shot. Which of the following conditions must be satisfied? (a) α > β (b) T cos α = (T − t) cos β (c) (T − t) cos α = T cos β (d) u sin α T − (1/2) g T^2 = u sin β (T − t) − (1/2) g (T − t)^2

$$\mathrm{Two}\:\mathrm{shells}\:\mathrm{are}\:\mathrm{fired}\:\mathrm{from}\:\mathrm{a}\:\mathrm{canon}\:\mathrm{with}\:\mathrm{speed}\:\mathrm{u}\:\mathrm{each},\:\mathrm{at} \\ $$$$\mathrm{angles}\:\mathrm{of}\:\alpha\:\mathrm{and}\:\beta\:\mathrm{respectively}\:\mathrm{with}\:\mathrm{the}\:\mathrm{horizontal}.\:\mathrm{The} \\ $$$$\mathrm{time}\:\mathrm{interval}\:\mathrm{between}\:\mathrm{the}\:\mathrm{shots}\:\mathrm{is}\:{t}.\:\mathrm{They}\:\mathrm{collide}\:\mathrm{in}\:\mathrm{mid} \\ $$$$\mathrm{air}\:\mathrm{after}\:\mathrm{time}\:{T}\:\mathrm{from}\:\mathrm{the}\:\mathrm{first}\:\mathrm{shot}.\:\mathrm{Which}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{following}\:\mathrm{conditions}\:\mathrm{must}\:\mathrm{be}\:\mathrm{satisfied}? \\ $$$$\left({a}\right)\:\alpha\:>\:\beta \\ $$$$\left({b}\right)\:{T}\:\mathrm{cos}\:\alpha\:=\:\left({T}\:−\:{t}\right)\:\mathrm{cos}\:\beta \\ $$$$\left({c}\right)\:\left({T}\:−\:{t}\right)\:\mathrm{cos}\:\alpha\:=\:{T}\:\mathrm{cos}\:\beta \\ $$$$\left({d}\right)\:{u}\:\mathrm{sin}\:\alpha\:{T}\:−\:\frac{\mathrm{1}}{\mathrm{2}}\:{g}\:{T}^{\mathrm{2}} \:=\:{u}\:\mathrm{sin}\:\beta\:\left({T}\:−\:{t}\right)\:−\:\frac{\mathrm{1}}{\mathrm{2}}\:{g}\:\left({T}\:−\:{t}\right)^{\mathrm{2}} \\ $$

Answered by ajfour last updated on 08/Sep/17

At time T x_α =x_β (ucos β)(T−t)=(ucos α)T ⇒ Tcos α = (T−t)cos β hence (a) need be satisfied also at time T, y_α =y_β hence (d) should also be satisfied.

$${At}\:\:{time}\:{T}\:\:\:\:{x}_{\alpha} ={x}_{\beta} \\ $$$$\left({u}\mathrm{cos}\:\beta\right)\left({T}−{t}\right)=\left({u}\mathrm{cos}\:\alpha\right){T} \\ $$$$\Rightarrow\:\:\:\:\:{T}\mathrm{cos}\:\alpha\:=\:\left({T}−{t}\right)\mathrm{cos}\:\beta\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{{hence}}\:\left(\boldsymbol{{a}}\right)\:{need}\:{be}\:{satisfied} \\ $$$${also}\:{at}\:{time}\:{T},\:\:{y}_{\alpha} ={y}_{\beta} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{{hence}}\:\left(\boldsymbol{{d}}\right)\:{should}\:{also}\:{be} \\ $$$${satisfied}. \\ $$

Commented by Tinkutara last updated on 08/Sep/17

Thank you very much Sir!

$$\mathrm{Thank}\:\mathrm{you}\:\mathrm{very}\:\mathrm{much}\:\mathrm{Sir}! \\ $$

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