Question

A projectile is required to hit a target whose coordinates relative horizontal and vertical axes through the point of projection are $(\alpha ,\beta )$. If the gun velocity is $\sqrt{2g\alpha }$, it is impossible to hit the target if

• A. $\beta >\frac{3}{4}\alpha$
• B. $\beta \ge \frac{3}{4}\alpha$
• C. $\beta \le \frac{3}{4}\alpha$
• D. $\beta <\frac{3}{4}\alpha$

Answer 1

The correct option is A $\beta >\frac{3}{4}\alpha$

Equation of trajectory $y=x\tan \theta -\frac{g{x}^2}{2u^2}{\sec }^2\theta$ and it passes through point $(\alpha ,\beta )$

$\beta =\alpha \tan \theta -\frac{g{\alpha }^2}{2\left(2g\alpha \right)}(1+{\tan }^2\theta )$

$\frac{\alpha }{4}{\tan }^2\theta -\alpha \tan \theta +(\beta +\frac{\alpha }{4})=0$

If the value of discriminant $△=B^2-4AC<0$ for above quadratic equation then it would be impossible for projectile to hit target

${\alpha }^2-4\left(\frac{\alpha }{4}\right)(\beta +\frac{\alpha }{4})<0\Rightarrow \beta >\frac{3\alpha }{4}$