Question

Two identical balls are projected, one vertically up and the other one at an angle of ${30^ \circ }$ to the horizontal with the same initial speed. The potential energy at the respective highest point will be in the ratio:
A) $3:4$
B) $4:3$
C) $4:1$
D) $1:4$

Answer 1

We know that Potential Energy (P.E.) at a height (vertical) from the ground is equal to $mgh$, where $m$ = mass of the object, $g$ = acceleration due to gravity and $h$ = vertical height from the ground. We also know from the equation of motion that $h = \dfrac{{{u^2}{{\sin }^2}\theta }}{{2g}}$, where $u$ = initial speed, $\theta$ = angle made with the ground.

Complete step by step answer:
For the first ball, P.E. = $mg{h_1}$, where ${h_1}$ = height from the ground at the highest point for the first ball.
Thus, ${h_1} = \dfrac{{{u_1}^2{{\sin }^2}{\theta _1}}}{{2g}}$, ${\theta _1}$ = angle with horizontal for first ball = ${90^ \circ }$
Similarly for the second ball, P.E. = $mg{h_2}$, where ${h_2}$ = height from the ground at the highest point for the second ball.
Thus ${h_2} = \dfrac{{{u_2}^2{{\sin }^2}{\theta _2}}}{{2g}}$, ${\theta _2}$ = angle with horizontal for second ball = ${30^ \circ }$

Therefore,
Since the initial speed and acceleration due to gravity is same for the two balls,
$P.E{._1}$ = potential energy of first ball at highest point
$P.E{._2}$ = potential energy of second ball at highest point
Their ratio $\dfrac{{P.E{._1}}}{{P.E{._2}}}$
= $\dfrac{{{h_1}}}{{{h_2}}}$
= $\dfrac{{{{\sin }^2}{\theta _1}}}{{{{\sin }^2}{\theta _2}}}$
= $\dfrac{{{{\sin }^2}{{90}^ \circ }}}{{{{\sin }^2}{{30}^ \circ }}}$
= $\dfrac{4}{1}$
= $4:1$
Therefore, the answer is option $C$.

Note: Be careful about calculation mistakes that might creep in while performing substitutions. Remember the equation of motion and the potential energy formula and how to use them together by eliminating the same variables. We also see that height is directly proportional to ${\sin ^2}\theta$ if the initial speed is the same. This direct result can be used in questions to calculate ratios of height by just taking ratio of sine of the angles. Potential Energy is always calculated for vertical height only since it depends only on the vertical height and not on the horizontal displacement as we can see in the question that one ball goes vertically up and other follows projectile motion (both vertical and horizontal), but only vertical displacement of both is taken into account to solve the problem.