Question

a projectile is thrown with an initial velocity of (Xi + Y j )metre per second if the range of the projectile is double the maximum height what is the relation between X and Y ​

Answer 1

Let $\mathsf{u=X}\ \mathbf{\hat i}\ \mathsf{+\ Y}\ \mathbf{\hat j}$ be the initial velocity and $\sf{\theta}$ be the angle of projection. Then we have,

$\quad$

$\sf{\tan\theta=\dfrac {Y}{X}}$

$\quad$

So, given,

$\quad$

$\sf{R=2H}\\\\\\\sf{\dfrac {u^2\cdot2\sin\theta\cos\theta}{g}=2\cdot\dfrac {u^2\sin^2\theta}{2g}}\\\\\\\sf{2\cos\theta=\sin\theta}\\\\\\\sf{\tan\theta=2}$

$\quad$

That is,

$\quad$

$\sf{\dfrac {Y}{X}=2}\\\\\\\sf{\underline {\underline {Y=2X}}}$