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Two bodies are projected from the same ...

Two bodies are projected from the same point with equal speeds in such directions that they both strike the same point on a plane whose inclination is `beta`. If `alpha` the angle of projection of the first body with the horizontal , show that the ratio of their time of flights is `(sin (alpha - beta))/(cos alpha)`

Text Solution

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Let `alpha` be the angle of projection of the second body
`R=(u^(2))/(g cos^(2)beta)[sin(2 alpha-beta)-sin beta]`
Range of both the bodies is same, therefore
`sin (2alpha-beta)=sin(2 alpha.-beta)` or `sin 2 alpha.-beta=pi-(2alpha-beta)`,

`alpha.=(pi)/2-(alpha-beta)`
Now `T=(2u sin (alpha-beta))/(g cos beta)` and `T.=(2u sin (alpha.-beta))/(g cos beta)`
`:.T/(T.)=(2sin (alpha-beta))/(2sin (alpha.-beta))=(sin(alpha-beta))/(sin{(pi)/2-(alpha-beta)-beta})`
`sin=((alpha-beta))/(sin((pi)/2-alpha))=(sin(alpha-beta))/(cos alpha)`
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