Question

Two stones are thrown vertically upwards simultaneously with their initial velocities $u_1$ and $u_2$ respectively. Calculate the ratio of heights reached by them.

Answer 1

Step 1: Free fall and motion

1. When any object is thrown vertically upwards, the acceleration for the object can be considered the same as the acceleration due to gravity.
2. Since the acceleration due to gravity works in a downwards direction, it can be considered negative in this case, as the stones are thrown upwards.
3. At maximum height, the final velocity of the stones becomes zero.

Step 2: Calculation of time for stone 1.

Calculation of the maximum height reached by stone 1.

${\mathrm{v}}_1^2={\mathrm{u}}_1^2-2{\mathrm{gs}}_1$

Here,

1. $v_1$ is the final velocity of the stone 1.
2. $u_1$ is the initial velocity of the stone 1.
3. $g$ is the acceleration due to gravity.
4. $s_1$ is the maximum height reached by a stone 1.

At maximum height, the final velocity is zero.

$$\begin{gathered} 0={\mathrm{u}}_1^2-2{\mathrm{gs}}_1 \\ 2{\mathrm{gs}}_1={\mathrm{u}}_1^2 \\ {\mathrm{s}}_1=\frac{{\mathrm{u}}_1^2}{2\mathrm{g}} \end{gathered}$$

Step 2: Calculation of time for stone 2.

Calculation of the maximum height reached by stone 2.

${\mathrm{v}}_2^2={\mathrm{u}}_2^2-2{\mathrm{gs}}_2$

Here,

1. $v_2$ is the final velocity of the stone 2.
2. $u_2$ is the initial velocity of the stone 2.
3. $s_2$ is the maximum height reached by stone 2.

At maximum height, the final velocity is zero.

$$\begin{gathered} 0={\mathrm{u}}_2^2-2{\mathrm{gs}}_2 \\ 2{\mathrm{gs}}_2={\mathrm{u}}_2^2 \\ {\mathrm{s}}_2=\frac{{\mathrm{u}}_2^2}{2\mathrm{g}} \end{gathered}$$

Step 3: Calculation of the ratio of the maximum heights:

$$\begin{gathered} \frac{{\mathrm{s}}_1}{{\mathrm{s}}_2}=\frac{{\displaystyle \frac{{\mathrm{u}}_1^2}{2\mathrm{g}}}}{\frac{{\mathrm{u}}_2^2}{2\mathrm{g}}} \\ \frac{{\mathrm{s}}_1}{{\mathrm{s}}_2}=\frac{{\mathrm{u}}_1^2}{{\mathrm{u}}_2^2} \end{gathered}$$

Hence, the ratio of the maximum height reached by the stones is ${\mathrm{u}}_1^2:{\mathrm{u}}_2^2$.