Two projectiles are fired at different angles with the same magnitude of velocity such that they have the same range. At what angles they might have been projected ?

Options

A. `35^(@) and 75^(@)`
B. `25 ^(@) and 65^(@)`
C. `10^(@)and 59^(@)`
D. None of these

Answer

To solve the problem of finding the angles at which two projectiles can be fired to achieve the same range with the same initial velocity, we can follow these steps:

### Step-by-Step Solution:

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

2. **Set Up the Problem**:
We know that both projectiles have the same initial velocity \( u \) and achieve the same range \( R \). Therefore, we can equate the range formulas for both angles:
\[ R_1 = R_2 \]
This implies:
\[ \frac{u^2 \sin(2\theta_1)}{g} = \frac{u^2 \sin(2\theta_2)}{g} \]
Since \( u \) and \( g \) are constants and non-zero, we can simplify this to:
\[ \sin(2\theta_1) = \sin(2\theta_2) \]

3. **Use the Property of Sine**:
The sine function has a property where:
\[ \sin(180^\circ - x) = \sin(x) \]
This means that if \( \sin(2\theta_1) = \sin(2\theta_2) \), then:
\[ 2\theta_2 = 180^\circ - 2\theta_1 \quad \text{or} \quad 2\theta_2 = 2\theta_1 \]

4. **Find the Relationship Between Angles**:
From the first equation, we can derive:
\[ 2\theta_2 = 180^\circ - 2\theta_1 \]
Dividing by 2 gives:
\[ \theta_2 = 90^\circ - \theta_1 \]
This shows that the two angles \( \theta_1 \) and \( \theta_2 \) are complementary angles.

5. **Conclusion**:
Therefore, the angles at which the projectiles might have been projected are complementary, meaning:
\[ \theta_1 + \theta_2 = 90^\circ \]

6. **Check the Options**:
Given options are:
- Option A: \( 35^\circ + 75^\circ = 110^\circ \) (Not valid)
- Option B: \( 25^\circ + 65^\circ = 90^\circ \) (Valid)
- Option C: \( 10^\circ + 59^\circ = 69^\circ \) (Not valid)

Thus, the correct answer is Option B: \( 25^\circ \) and \( 65^\circ \).