Question

Pankaj and Sudhir are playing with two different balls of masses m and 2m, respectively. If Pankaj throws his ball vertically up and Sudhir at an angle, both the balls stay in our view for the same period. Find the ratio of the heights attained by the two balls.

Answer 1

In this question we will use the expression of time of flight and height attained by the two balls. Further, we will study the basics of time of flight. Also, we will study about the potential and kinetic energy which comes into account when an object is at a certain height.
Formula used:
$T = \dfrac{{2u}}{g}$
$H = \dfrac{{u_{}^2}}{{2g}}$

Complete answer:
As we know that time of flight is defined as the measurement of the time taken by an object or particle to travel a distance through a medium.
So, the time of flight of the ball thrown by Pankaj is given by:
${T_1} = \dfrac{{2{u_1}}}{g}$
Time of flight of the ball thrown by Sudhir is given by:
${T_2} = \dfrac{{2{u_2}\sin ({{90}^ \circ } - \theta )}}{g} = \dfrac{{2{u_2}\cos \theta }}{g}$
Here, we have;
\eqalign{ & {T_1} = {T_2} \cr & \Rightarrow \dfrac{{2{u_1}}}{g} = \dfrac{{2{u_2}\cos \theta }}{g} \cr & \therefore {u_1} = {u_2}\cos \theta \cr}
Height of the ball thrown by Pankaj is given by:
${H_1} = \dfrac{{u_1^2}}{{2g}}$
Height of the ball thrown by Sudhir is given by:
\eqalign{& {H_2} = \dfrac{{u_2^2{{\sin }^2}({{90}^ \circ } - \theta )}}{{2g}} \cr & \Rightarrow {H_2} = \dfrac{{u_2^2{{\cos }^2}\theta }}{{2g}} \cr}
$\therefore \dfrac{{{H_1}}}{{{H_2}}} = \dfrac{{\dfrac{{u_1^2}}{{2g}}}}{{\dfrac{{u_2^2{{\cos }^2}\theta }}{{2g}}}} = 1$
Therefore, the ratio of the heights attained by the two balls is 1.

Additional information:
As we know that when an object or particle is at a certain height, it gains potential energy.
Further, 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.
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:
Here, we should remember that time of flight can be measured for waves as well. Total energy is given by the sum of potential energy and kinetic energy. The unit of energy here is electron volt which is $1.602 \times {10^{ - 19}}Joule$. The S.I unit of energy is Joule. At the maximum height the potential energy is maximum and kinetic energy is zero