Question

A particle is projected with a velocity v such that its range on a horizontal plane is twice the greatest height attained by it. The range of the projectile will be
(g is the acceleration due to gravity).

Answer 1

In this question we will use the relation between height and range of projectile. By putting the given values and solving the equation, we will get the required result. Also, we will discuss the basics of mass conservation law, collision when particles collide and also about the effect on energy when a particle reaches a certain height.
Formula used:
$R = 4H\cot \theta$

Complete answer:
Here, the range R is given by:
$R = 4H\cot \theta$
Now, putting the value of R in above equation we get:
$\cot \theta = \dfrac{1}{2}$
Now, from the value of $\cot \theta$, we can say that:
$\sin \theta = \dfrac{2}{{\sqrt 5 }}$
$\cos \theta = \dfrac{1}{{\sqrt 5 }}$
Now, we will use these values to find the range of the projectile:
$R = \dfrac{{2{v^2}\sin \theta \cos \theta }}{g}$
By substituting the values in this equation we get:
\eqalign{& \Rightarrow R = \dfrac{{2{v^2}}}{g} \times \dfrac{2}{{\sqrt 5 }} \times \dfrac{1}{{\sqrt 5 }} \cr & \therefore R = \dfrac{{4{v^2}}}{{5g}} \cr}
Therefore, we get the required result, which tells the range of the projectile of the given particle.

Additional information:
From the law of conservation of mass which says that mass can neither be created nor destroyed in any chemical reactions. Also, it can be defined as the mass of any one element at the beginning of a reaction will equal the mass of that element at the end of the reaction.
We know that collision means when two objects come in contact with each other for a very short period. Collision is an interaction between two masses for a very short interval where the momentum and energy of the colliding masses changes. Here are two types of collision: first is elastic collision where the energy remains the same after interaction or collision and second is inelastic collision where the final energy changes after the collision of the particles or body.
As we know that energy is defined as the ability to do work. Energy can be found in many things and can take different forms, like-kinetic energy is the energy of motion, and potential energy is energy due to an object's position or structure. Total energy of a system is given by the sum of kinetic energy and potential energy.
We also know that, according to the law of conservation of energy, energy can neither be created nor be destroyed; rather it can only be transferred from one form to another. Here, in this question electrical energy is transferred to heat energy.

Note:
we should remember, during the solution, one should see that laws of conservation are not violated, i.e., mass can neither be created nor be destroyed and similarly for energy of the particles as well. We should know that, when a particle reaches maximum height, the kinetic energy of the particle is zero and potential energy is maximum.