Question

Two stones have different masses $m_1$ and $m_2$ are projected at angles $\theta$ and $\left(90-\theta \right)$ with the same speed from the same point. Calculate the ratio of their maximum heights.

Answer 1

Step 1: Formula used for the maximum height reached due to projectile motion

The maximum height reached by the stones can be expressed as follows:

$\mathrm{H}=\frac{{\mathrm{u}}^2{\sin }^2\mathrm{\theta }}{2\mathrm{g}}$

Here,

1. $\mathrm{H}$ is the maximum height reached.
2. $\mathrm{u}$ is the initial velocity.
3. $\mathrm{\theta }$ is the angle at which the stone is thrown.
4. $\mathrm{g}$ is the acceleration due to gravity.

2. The maximum height reached by the stones does not depend upon the masses of the stones.

Step 2: Calculation of maximum height for stone 1

Calculate the maximum height reached by the stone 1.

${\mathrm{H}}_1=\frac{{\mathrm{u}}^2{\sin }^2\mathrm{\theta }}{2\mathrm{g}}$

Here,

${\mathrm{H}}_1$ is the maximum height reached by the stone $1$.

Step 2: Calculation of maximum height for stone 2

Calculate the maximum height reached by the stone 2.

${\mathrm{H}}_2=\frac{{\mathrm{u}}^2{\sin }^2\left(90-\mathrm{\theta }\right)}{2\mathrm{g}}$

Here,

${\mathrm{H}}_2$ is the maximum height reached by the stone $2$.

Step 3: Calculate the ratio of heights

Calculate the ratio of the heights for both stones.

$$\begin{gathered} \frac{{\mathrm{H}}_1}{{\mathrm{H}}_2}=\frac{{\displaystyle \frac{{\mathrm{u}}^2{\sin }^2\mathrm{\theta }}{2\mathrm{g}}}}{{\displaystyle \frac{{\mathrm{u}}^2{\sin }^2\left(90-\mathrm{\theta }\right)}{2\mathrm{g}}}} \\ \frac{{\mathrm{H}}_1}{{\mathrm{H}}_2}=\frac{{\sin }^2\mathrm{\theta }}{{\cos }^2\mathrm{\theta }} \\ \frac{{\mathrm{H}}_1}{{\mathrm{H}}_2}=\frac{{\tan }^2\mathrm{\theta }}{1} \end{gathered}$$

Hence, the ratio of the maximum heights for both stones is ${\tan }^2\mathrm{\theta }:1$.