Two projectiles `A` and `B` are thrown from the same point with velocities `v` and `v/2` respectively. If `B` is thrown at an angle `45^(@)` with horizontal.What is the inclination of `A`.when their ranges are the same?

Options

A. `sin^(-1) ((1)/(4))`
B. `(1)/(2)sin^(-1) ((1)/(4))`
C. `2 sin^(-1) ((1)/(4))`
D. `(1)/(2) sin^(-1)((1)/(8))`

Explanation

To solve the problem, we need to find the inclination angle of projectile A when the ranges of projectiles A and B are the same. Let's break down the steps:

### Step 1: Understand the given information
- Projectile A is thrown with a velocity \( v \).
- Projectile B is thrown with a velocity \( \frac{v}{2} \) at an angle of \( 45^\circ \) with the horizontal.
- We need to find the angle of inclination \( \theta_A \) for projectile A such that the ranges of both projectiles are equal.

### Step 2: Write the formula for the range of a projectile
The range \( R \) of a projectile thrown with an initial velocity \( u \) at an angle \( \alpha \) is given by:
\[ R = \frac{u^2 \sin(2\alpha)}{g} \]
where \( g \) is the acceleration due to gravity.

### Step 3: Calculate the range of projectile B
For projectile B:
- Initial velocity \( u_B = \frac{v}{2} \)
- Angle \( \alpha_B = 45^\circ \)

Using the range formula:
\[ R_B = \frac{\left(\frac{v}{2}\right)^2 \sin(2 \times 45^\circ)}{g} \]
Since \( \sin(90^\circ) = 1 \):
\[ R_B = \frac{\frac{v^2}{4} \cdot 1}{g} = \frac{v^2}{4g} \]

### Step 4: Set up the equation for the range of projectile A
For projectile A:
- Initial velocity \( u_A = v \)
- Angle \( \alpha_A = \theta_A \)

Using the range formula:
\[ R_A = \frac{v^2 \sin(2\theta_A)}{g} \]

### Step 5: Equate the ranges of projectiles A and B
Since the ranges are the same:
\[ R_A = R_B \]
\[ \frac{v^2 \sin(2\theta_A)}{g} = \frac{v^2}{4g} \]

### Step 6: Simplify the equation
Cancel \( v^2 \) and \( g \) from both sides:
\[ \sin(2\theta_A) = \frac{1}{4} \]

### Step 7: Solve for \( \theta_A \)
To find \( 2\theta_A \):
\[ 2\theta_A = \sin^{-1}\left(\frac{1}{4}\right) \]
Now, divide by 2 to find \( \theta_A \):
\[ \theta_A = \frac{1}{2} \sin^{-1}\left(\frac{1}{4}\right) \]

### Final Answer
The inclination of projectile A is:
\[ \theta_A = \frac{1}{2} \sin^{-1}\left(\frac{1}{4}\right) \]

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