Question

How do you determine the maximum height of a projectile?

Answer 1

In this question, we will first use the expression which gives the relation between maximum height, vertical and horizontal component of projectile. Also, by Substituting the values in this expression will help us to get the required result of maximum height of the projectile.
Formula used:
$v_{{h_{\max }}}^2 = v_{0y}^2 - 2.g.{h_{\max }}$

Complete answer:
When we launch a projectile at an angle $\theta$ from the horizontal, the initial velocity of the projectile will have a vertical and a horizontal component given as:
\eqalign{& {v_{0x}} = {v_0}.\cos \theta \cr & {v_{0y}} = {v_0}.\sin \theta \cr}
Now, we have to determine the maximum height that is reached by the projectile during its flight, so, here we need to take a look at the vertical component of the body’s motion.

As we know that vertically, the motion of the projectile is affected by the gravity. This shows that at the maximum height, the vertical component of the initial speed of the projectile will be zero.
Now the projectile will decelerate on its way to maximum height, this will come to a complete stop at maximum height and then it will start its free fall towards the ground or surface.
Here, if we use the vertical component of the initial speed, we have:
$v_{{h_{\max }}}^2 = v_{0y}^2 - 2.g.{h_{\max }}$
Here,$v_{{h_{\max }}}^2 = 0$, so the above equation becomes:
$v_{0y}^2 = 2.g.{h_{\max }}$
So, the maximum height that is reached by the projectile will be given by:
${h_{\max }} = \dfrac{{v_{0y}^2}}{{2.g}}$
$\therefore {h_{\max }} = \dfrac{{v_0^2.{{\sin }^2}\theta }}{{2.g}}$
Therefore we get the required answer.

Note:
We should remember that when an object's gains maximum height, then only potential energy exists, whereas kinetic energy exists when the object gains motion. We should remember that the laws of conservation of energy can never be violated in any case. 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.