Question

A ball of mass m is thrown vertically upwards. Another ball of mass 2m is thrown up making an angle θ with the vertical. Both of them stay in air for the same time. Their maximum heights are in the ratio of:
A: 2: 1
B: 1: 1
C: $1:\cos \theta $
D: $1:\sec \theta $

Answer 1

Hint:When such questions are given, the equations of kinematics that relates all the given quantities are to be used to solve the questions. The equations of projectile motion that connects the given quantities like time, mass, height, angle etc can be used to solve this particular question.

Formulas used:Time of flight: $T=\dfrac{2u\sin \theta }{g}$
Maximum height: $H=\dfrac{{{u}^{2}}{{\sin }^{2}}\theta }{g}$, where u is the initial velocity with which the body is projected and $\theta $is the angle that it makes with the horizontal.

Complete step by step answer:
We are given two balls that are projected at different instances. When the first ball of mass m is projected vertically upwards, the second ball of mass 2m makes an angle $\theta $ with the vertical.

The time of flight is the same for both the balls.
$\begin{align}
& {{T}_{1}}=\dfrac{2{{u}_{1}}}{g} \\
& {{T}_{2}}=\dfrac{2{{u}_{2}}\sin \theta }{g} \\
& \Rightarrow \dfrac{2{{u}_{1}}}{g}=\dfrac{2{{u}_{2}}\sin \theta }{g} \\
& \Rightarrow {{u}_{1}}={{u}_{2}}\sin \theta ..........(1) \\
& H=\dfrac{{{u}^{2}}{{\sin }^{2}}\theta }{2g} \\
& \Rightarrow \dfrac{{{H}_{1}}}{{{H}_{2}}}=\dfrac{{{u}_{1}}^{2}}{u_{2}^{2}{{\sin }^{2}}\theta }......(2)
\\
& 1\And 2\Rightarrow \\
& \dfrac{{{H}_{1}}}{{{H}_{2}}}=1 \\
\end{align}$
Hence, the maximum heights of the balls are at a ratio of 1:1.

Thus, we can conclude that option B is the correct answer among the given four options.

Note:The time taken to reach the maximum height by a projectile is the time that it requires to reach the topmost position of the trajectory. The time of flight is the overall time taken by the projectile to remain in air that is the time between throwing and landing.Both these times can be related as the time of flight is double the time required by the body to reach the maximum height of the trajectory.