Question

A particle is projected with a velocity $u$ making an angle $\theta$ with the horizontal such that the trajectory just grazes the vertices of the triangle then

• A. $\tan \theta =\cos \alpha +\cos \beta$
• B. $\cot \theta =\cos \alpha +\cos \beta$
• C. $\sin \theta =\sin \alpha +\sin \beta$
• D. $\tan \theta =\tan \alpha +\tan \beta$

Answer 1

Answer: (D)

$\tan \theta =\tan \alpha +\tan \beta$

Let the ball touch the upper vertex at length $r_1$ from angle $\alpha$ along XZ and $r_2$ from angle $\beta$ along ZX.
When it touches upper vertex, height $h=r_1\tan \alpha =r_2\tan \beta$.
Use $y=r_1\tan \alpha$ and $x=r_1$ in the equation of path of a projectile to find $r_1$.
Use:
Range $R=r_1+r_2$ and $\frac{r_1}{r_2}=\frac{\tan \beta }{\tan \alpha }$ along with value of $r_1=\frac{2u^2{\cos }^2\theta }{g}(\tan \theta -\tan \alpha )$ from above to get

$\tan \theta =\tan \alpha +\tan \beta$