An arrow is shot in air, its time of flight is 5 sec and horizontal range is 200 m. the inclination of the arrow with the horizontal is

Options

A. `tan^(-1).(5)/(8)`
B. `tan^(-1).(8)/(5)`
C. `tan^(-1).(1)/(8)`
D. `45^(@)`

Explanation

To find the inclination of the arrow with the horizontal, we can use the information given: time of flight (T) is 5 seconds and horizontal range (R) is 200 meters. We will use the equations of projectile motion to derive the angle of projection (θ).

### Step-by-Step Solution:

1. **Understand the relationship between time of flight, horizontal range, and initial velocity:**
The horizontal range (R) of a projectile is given by the formula:
\[ R = V \cdot \cos(\theta) \cdot T \]
where \(V\) is the initial velocity, \(\theta\) is the angle of projection, and \(T\) is the time of flight.

2. **Substitute the known values into the range formula:**
Given \(R = 200 \, \text{m}\) and \(T = 5 \, \text{s}\), we can write:
\[ 200 = V \cdot \cos(\theta) \cdot 5 \]
Simplifying this gives:
\[ V \cdot \cos(\theta) = \frac{200}{5} = 40 \, \text{m/s} \]

3. **Use the time of flight to find the vertical component of the initial velocity:**
The time of flight is also related to the vertical component of the initial velocity (\(V \cdot \sin(\theta)\)) by the formula:
\[ T = \frac{2V \cdot \sin(\theta)}{g} \]
where \(g\) is the acceleration due to gravity (approximately \(9.81 \, \text{m/s}^2\)). Rearranging gives:
\[ V \cdot \sin(\theta) = \frac{g \cdot T}{2} = \frac{9.81 \cdot 5}{2} = 24.525 \, \text{m/s} \]

4. **Now we have two equations:**
From step 2:
\[ V \cdot \cos(\theta) = 40 \, \text{m/s} \quad (1) \]
From step 3:
\[ V \cdot \sin(\theta) = 24.525 \, \text{m/s} \quad (2) \]

5. **Divide equation (2) by equation (1) to find \(\tan(\theta)\):**
\[ \frac{V \cdot \sin(\theta)}{V \cdot \cos(\theta)} = \frac{24.525}{40} \]
This simplifies to:
\[ \tan(\theta) = \frac{24.525}{40} = 0.613125 \]

6. **Find the angle \(\theta\):**
To find \(\theta\), take the arctangent:
\[ \theta = \tan^{-1}(0.613125) \]
Using a calculator, we find:
\[ \theta \approx 31.0^\circ \]

### Final Answer:
The inclination of the arrow with the horizontal is approximately \(31.0^\circ\).