Two stones are thrown with same speed u at different angles from ground n air if both stones have same range and height attained by them are `h_(1) and h_(2)`, then `h_(1) + h_(2)` is equal to

Options

A. `(u^(2))/(g)`
B. `(u^(2))/(2g)`
C. `(u^(2))/(3g)`
D. `(u^(2))/(4g)`

Answer

To solve the problem, we need to find the sum of the heights attained by two stones thrown with the same speed \( u \) at different angles, given that they have the same range. Let's denote the heights attained by the stones as \( h_1 \) and \( h_2 \).

### Step 1: Understand the Range and Height Formulas
The range \( R \) of a projectile is given by the formula:
\[ R = \frac{u^2 \sin 2\theta}{g} \]
where \( \theta \) is the angle of projection, \( u \) is the initial speed, and \( g \) is the acceleration due to gravity.

The maximum height \( h \) attained by the projectile is given by:
\[ h = \frac{u^2 \sin^2 \theta}{2g} \]

### Step 2: Set Up the Problem
Let the angles of projection for the two stones be \( \theta_1 \) and \( \theta_2 \). Since both stones have the same range, we can write:
\[ \frac{u^2 \sin 2\theta_1}{g} = \frac{u^2 \sin 2\theta_2}{g} \]
This simplifies to:
\[ \sin 2\theta_1 = \sin 2\theta_2 \]

### Step 3: Analyze the Sine Function
The equation \( \sin 2\theta_1 = \sin 2\theta_2 \) implies that:
\[ 2\theta_1 = 2\theta_2 + n\pi \quad \text{or} \quad 2\theta_1 = \pi - 2\theta_2 + n\pi \]
for some integer \( n \). For angles in the range of projectile motion, we can conclude that:
\[ \theta_1 + \theta_2 = 90^\circ \]
This means \( \theta_2 = 90^\circ - \theta_1 \).

### Step 4: Calculate Heights
Now we can calculate the heights \( h_1 \) and \( h_2 \):
- For \( h_1 \):
\[ h_1 = \frac{u^2 \sin^2 \theta_1}{2g} \]
- For \( h_2 \):
\[ h_2 = \frac{u^2 \sin^2 \theta_2}{2g} = \frac{u^2 \sin^2 (90^\circ - \theta_1)}{2g} = \frac{u^2 \cos^2 \theta_1}{2g} \]

### Step 5: Sum the Heights
Now we can find \( h_1 + h_2 \):
\[ h_1 + h_2 = \frac{u^2 \sin^2 \theta_1}{2g} + \frac{u^2 \cos^2 \theta_1}{2g} \]
Factoring out \( \frac{u^2}{2g} \):
\[ h_1 + h_2 = \frac{u^2}{2g} (\sin^2 \theta_1 + \cos^2 \theta_1) \]
Using the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \):
\[ h_1 + h_2 = \frac{u^2}{2g} \cdot 1 = \frac{u^2}{2g} \]

### Final Answer
Thus, the sum of the heights attained by the two stones is:
\[ h_1 + h_2 = \frac{u^2}{2g} \]