Two balls are projected from the same point in the direction `30^(@)` and `60^(@)` with respect to the horizontal. What is the ratio of their initial velocities if they (a) attain the same height (b) have the same range?

Explanation

`theta_(1)=60^(@), theta_(2)=30^(@)`
First body maximum height `H_(1)`
`=(u^(2)sin^(2)theta)/(2g)=(u_(1)^(2)sin^(2)60^(@))/(2g)`
Second body maximum height `H_(2)`
`=(u^(2)sin^(2)theta)/(2g)=(u^(2)sin^(2)30^(@))/(2g)" But "H_(1)-H_(2)`
`therefore u_(1)^(2)/u_(2)^(2)=(sin^(2)60^(@))/(sin^(2)30^(@))=(3/4)/(1/4)=3`
`u_(1):u_(2)=sqrt(3):1" (Or) "3:1`
If both the bodies have same range `R_(1)=R_(2)`
`(u_(1)^(2)sin2 times 60)/g=(u_(2)^(2)sin2 times 30)/g`
`rArr u_(1)^(2)/u_(2)^(2)=(sin60^(@))/(sin120^(@))=(sin60^(@))/(cos30^(@))=(sqrt(3)//2)/(sqrt(3)//2)=1`
`u_(1):u_(2)=1:1`