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A particle is projected over a traingle ...

A particle is projected over a traingle from one end of a horizontal base and grazing the vertex falls on the other end of the base. If `alpha` and `beta` be the base angles and `theta` the angle of projection, prove that `tan theta = tan alpha + tan beta`.

Text Solution

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If R is the range of the particle, then from the figure we have
`tanprop+tanbeta=y/x+y/(R-x)=(y(R-x)+xy)/(x(R-x))`
or `tanprop+tanbeta=y/xxxR/((R-y))`

Also, the trajectory of the particle is
`y=x tan theta[1-x/R]`
or `tantheta=y/x xx R/((R-x))`
From equations (1) and (2), we get `tantheta=tanprop+tanbeta`
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