Two stones having different masses `m_(1)` and `m_(2)` are projected at an angle `alpha` and `(90^(@) - alpha)` with same speed from same point. The ratio of their maximum heights is

Options

A. `1:1`
B. `1: tan alpha`
C. `tan alpha :1`
D. `tan^(2) alpha :1`

Explanation

To solve the problem of finding the ratio of the maximum heights attained by two stones projected at different angles, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have two stones projected from the same point with the same speed \( u_0 \). One stone is projected at an angle \( \alpha \) and the other at an angle \( 90^\circ - \alpha \).

2. **Formula for Maximum Height**:
The maximum height \( H \) achieved by a projectile is given by the formula:
\[ H = \frac{u_0^2 \sin^2 \theta}{2g} \]
where \( \theta \) is the angle of projection, \( g \) is the acceleration due to gravity, and \( u_0 \) is the initial speed.

3. **Calculate Maximum Height for Stone 1**:
For the first stone projected at angle \( \alpha \):
\[ H_1 = \frac{u_0^2 \sin^2 \alpha}{2g} \]

4. **Calculate Maximum Height for Stone 2**:
For the second stone projected at angle \( 90^\circ - \alpha \):
\[ H_2 = \frac{u_0^2 \sin^2 (90^\circ - \alpha)}{2g} \]
Using the trigonometric identity \( \sin(90^\circ - \theta) = \cos \theta \), we can rewrite this as:
\[ H_2 = \frac{u_0^2 \cos^2 \alpha}{2g} \]

5. **Finding the Ratio of Maximum Heights**:
Now, we need to find the ratio \( \frac{H_1}{H_2} \):
\[ \frac{H_1}{H_2} = \frac{\frac{u_0^2 \sin^2 \alpha}{2g}}{\frac{u_0^2 \cos^2 \alpha}{2g}} \]
The \( u_0^2 \) and \( 2g \) terms cancel out:
\[ \frac{H_1}{H_2} = \frac{\sin^2 \alpha}{\cos^2 \alpha} \]
This simplifies to:
\[ \frac{H_1}{H_2} = \tan^2 \alpha \]

6. **Final Ratio**:
Therefore, the ratio of the maximum heights \( H_1 : H_2 \) can be expressed as:
\[ H_1 : H_2 = \tan^2 \alpha : 1 \]

### Conclusion:
The ratio of the maximum heights attained by the two stones is \( \tan^2 \alpha : 1 \).