Two particles `A and B` are projected from the same point in different directions in such a manner that vertical components of their initial velocities are same (Fig. 5.8). Find the ratio of range.
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Explanation

Range of the projectile , in terms of velocity components can be written as followings:
`R=(2u_(x)u_(y))/(g)`
Let `u_(1) and u_(2)` be the initial speed of projection for P and Q respectively .
`R_(p)(2(u_(1) cos alpha)(u_(1) sin alpha))/(g)`
`R_(Q)=(2(u_(2) cos beta)(u_(2) sin beta))/(g)`
It given that `u_(1) sin alpha =u_(2) sin beta`, hence we can divide the above relations to get the following :
`(R_(p))/(R_(Q))=(u_(1))/(u^(2))xx(cos alpha)/(cos beta)`
Future we can substitute `(u_(1))/(u_(2))=(sin beta)/(sin alpha)`
`rArr (R_(p))/(R_(Q))=(sin beta)/(sin alpha)xx(xos alpha)/( cos beta)`
`rArr (R_(p))/(R_(Q))=(tan beta)/(tan alpha)`