Question

The maximum height reached by the projectile is 4m. The horizontal range is 12m. Velocity of projection in m/s is (g - acceleration due to gravity).
$
  {\text{A}}{\text{. }}5\sqrt {\dfrac{g}{2}} \\
  {\text{B}}{\text{. }}3\sqrt {\dfrac{g}{2}} \\
  {\text{C}}{\text{. }}\dfrac{1}{3}\sqrt {\dfrac{g}{2}} \\
  {\text{D}}{\text{. }}\dfrac{1}{5}\sqrt {\dfrac{g}{2}} \\
 $

Answer 1

Hint: Here we go through by applying the formula of projectile that we study in the chapter of kinematics. We use the formula of maximum height and the formula of range.
Formula used: - ${H_{\max }} = \dfrac{{{u^2}{{\sin }^2}\theta }}{{2g}}$ , $R = \dfrac{{{u^2}\sin 2\theta }}{g}$ where, u= initial velocity, $\theta = $angle made by initial velocity and horizontal line, g= acceleration due to gravity.

Complete Step-by-Step solution:
Here in the question it is given that the maximum height reached by the projectile is 4m and the horizontal range is 12 m. we have to find out the initial velocity i.e. u.

As we know that in a projectile projected at angle θ maximum height and range is given by:
${H_{\max }} = \dfrac{{{u^2}{{\sin }^2}\theta }}{{2g}}$ And $R = \dfrac{{{u^2}\sin 2\theta }}{g}$
By putting the given value in the formula we get,
 $4 = \dfrac{{{u^2}{{\sin }^2}\theta }}{{2g}}$………….. Let it be equation (1).
And $12 = \dfrac{{{u^2}\sin 2\theta }}{g}$………… let it be equation (2).
Now, dividing equation (2) by equation (1):
$ \Rightarrow 3 = \dfrac{{\sin 2\theta }}{{{{\sin }^2}\theta }} \times 2$
$ \Rightarrow \dfrac{3}{2} = \dfrac{{2\sin \theta \cos \theta }}{{{{\sin }^2}\theta }}$ As we know that ($\sin 2\theta = 2\sin \theta \cos \theta $)
$ \Rightarrow \tan \theta = \dfrac{4}{3}$ As we know$\dfrac{{\sin \theta }}{{\cos \theta }} = \tan \theta $.
And by using Pythagoras theorem that we learn in math we will find out the value of$\sin \theta $.
$ \Rightarrow \sin \theta = \dfrac{4}{5}$.
Now put the value of $\sin \theta $ in equation (1).
$
   \Rightarrow 4 = \dfrac{{{u^2}}}{{2g}} \times {\left( {\dfrac{4}{5}} \right)^2} \\
   \Rightarrow {u^2} = \dfrac{{25g}}{2} \\
  \therefore u = 5\sqrt {\dfrac{g}{2}} \\
 $
Hence option A is the correct answer.

Note: Whenever we face such a type of question the key concept for solving the question is to first find out the formula in which we can apply the given data of the question. And then by applying the formula we will make the equations and after solving the equation we will be able to find out our answers.