It was calculated that a shell when fired from a gun with a certain velocity and at an angle of elevation `5pi//36` radius should strike a given target. In actual practice it was found that a hill just intervened in the trajectory. At what angle of elevation should the gun be fired to hit the target ?

Options

A. `(5pi)/36rad`
B. `(11pi)/36rad`
C. `(7pi)/36 rad`
D. `(13pi)/36rad`

Answer

To solve the problem of determining the new angle of elevation at which the gun should be fired to hit the target while avoiding the intervening hill, we can follow these steps:

### Step 1: Understand the Initial Conditions
We know that the shell was initially fired at an angle of elevation \( \theta_1 = \frac{5\pi}{36} \) radians. The range \( R \) of the 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.

### Step 2: Determine the New Angle of Elevation
To avoid the hill and still hit the same target, we need to find a new angle of elevation \( \theta_2 \) such that the range remains the same. According to the properties of projectile motion, if one angle is \( \theta \), the complementary angle \( 90^\circ - \theta \) (or \( \frac{\pi}{2} - \theta \) in radians) will yield the same range.

Thus, we can express the new angle as:
\[ \theta_2 = \frac{\pi}{2} - \theta_1 \]

### Step 3: Substitute the Initial Angle
Substituting the value of \( \theta_1 \):
\[ \theta_2 = \frac{\pi}{2} - \frac{5\pi}{36} \]

### Step 4: Simplify the Expression
To simplify \( \theta_2 \), we need a common denominator:
\[ \theta_2 = \frac{18\pi}{36} - \frac{5\pi}{36} = \frac{(18 - 5)\pi}{36} = \frac{13\pi}{36} \]

### Step 5: Conclusion
Thus, the angle of elevation at which the gun should be fired to hit the target while avoiding the hill is:
\[ \theta_2 = \frac{13\pi}{36} \text{ radians} \]