Two bodies are projected from the same point with equal speeds in such directions that they both strikes the same point on a plane whose inclination is β. If a be the angle of projection of the first body with the horizontal, the ratio of their times of flight is

Question

Two bodies are projected from the same point with equal speeds in such directions that they both strikes the same point on a plane whose inclination is β. If a be the angle of projection of the first body with the horizontal, the ratio of their times of flight is

Infinity Learn · Accepted Answer

See fig. (3) Let α be the angle of projection of the second body. The range of both the bodies will be the same. Therefore, $$R=u2gcos2$$⁡β[$$sin$$⁡(2$$α−β$$)−$$sin$$⁡β] or  $$sin$$⁡(2$$α−β$$)=$$$$sin$$⁡$$2$$α′−β or  $$2$$α′−β$$=$$π$$=(2$$α−β$$) Now,  $$T=2usin$$⁡$$($$α−β$$)gcos$$⁡β and T′$$=2usin$$⁡α′−βgcos⁡β∴ TT′$$=$$$$sin$$⁡$$($$α−β$$)$$$$sin$$⁡α′−β$$=$$$$sin$$⁡$$($$α−β$$)$$$$sin$$⁡π$$2$$−$$($$α−β$$)$$−β$$=$$$$sin$$⁡$$($$α−β$$)$$$$sin$$⁡π$$2$$−α$$=$$$$sin$$⁡$$($$α−β$$)$$$$cos$$⁡α

Two bodies are projected from the same point with equal speeds in such directions that they both strikes the same point on a plane whose inclination is $β .$ If a be the angle of projection of the first body with the horizontal, the ratio of their times of flight is

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