A particle is moving in $$XY-plane$$. It starts to move from origin O at an angle a with $$X-axis$$. It has a constant acceleration in negative $$Y-direction$$. After some time it passes through a point B in a direction making angle β with the $$X-axis$$. If OB makes θ$$=45$$∘ angle with the $$X-axis$$, the value of $$tan$$α$$+$$$$tan$$⁡β is $$__________$$.

Question

A particle is moving in $$XY-plane$$. It starts to move from origin O at an angle a with $$X-axis$$. It has a constant acceleration in negative $$Y-direction$$. After some time it passes through a point B in a direction making angle β with the $$X-axis$$. If OB makes θ$$=45$$∘ angle with the $$X-axis$$, the value of $$tan$$α$$+$$$$tan$$⁡β is $$__________$$.

Infinity Learn · Accepted Answer

The $$X-coordinate$$ of point B $$isx=V0cos$$⁡αt  where, $$V0$$ is initial velocity and $$y=v0tsin$$⁡α−$$12a0t2$$.where, $$a0$$ is the acceleration Also, $$tan$$⁡θ$$=yxLet$$ the velocity of the particle is v at point B∴     vcos⁡β$$=v0cos$$⁡α and     vsin⁡β$$=v0sin$$⁡α−$$a0t$$∴     $$tan$$⁡β$$=vyvx=v0sin$$⁡α−$$a0tv0cos$$⁡α$$=$$$$tan$$⁡α−$$a0tv0cos$$⁡α   .....(i)and$$ $$tan$$⁡θ$$=yx=v0tsin$$⁡α−$$12a0t2v0tcos$$⁡αor $$2tan$$⁡θ$$=2tan$$⁡α−$$a0tv0cos$$⁡α    ......$$(ii)From$$ Eqs. $$(i) and (ii), we $$get2tan$$⁡θ−$$tan$$⁡β$$=2tan$$⁡α−$$tan$$⁡αor   $$2tan$$⁡θ$$=$$$$tan$$⁡α$$+$$$$tan$$⁡β∴$$tan$$⁡α$$+$$$$tan$$⁡β$$=2tan$$⁡$$45$$∘$$=2$$

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