An aeroplane is flying at a constant height of 1960 m with speed `600 kmh^(-1)` above the ground towards point directly over a person struggling in flood water. At what angle of sight with the vertical should be pilot release a survival kit if it is to reach the person in water ? `(g = 9.8 ms^(-2))`

Options

A. `45^(@)`
B. `30^(@)`
C. `60^(@)`
D. `90^(@)`

Explanation

To solve the problem of determining the angle at which the pilot should release a survival kit from an airplane flying at a height of 1960 m, we can follow these steps:

### Step 1: Convert the speed of the airplane from km/h to m/s
The speed of the airplane is given as 600 km/h. To convert this to meters per second (m/s), we use the conversion factor \( \frac{5}{18} \).

\[ \text{Speed in m/s} = 600 \times \frac{5}{18} = \frac{3000}{18} = \frac{500}{3} \text{ m/s} \]

### Step 2: Calculate the time taken for the survival kit to reach the ground
The time \( T \) taken for the survival kit to fall from the height \( h \) can be calculated using the formula for free fall:

\[ T = \sqrt{\frac{2h}{g}} \]

Where:
- \( h = 1960 \) m (height of the airplane)
- \( g = 9.8 \) m/s² (acceleration due to gravity)

Substituting the values:

\[ T = \sqrt{\frac{2 \times 1960}{9.8}} = \sqrt{\frac{3920}{9.8}} \approx \sqrt{400} = 20 \text{ seconds} \]

### Step 3: Calculate the horizontal distance (range) the survival kit travels
The horizontal distance \( R \) that the survival kit travels while it falls can be calculated using the formula:

\[ R = \text{speed} \times T \]

Substituting the values:

\[ R = \left(\frac{500}{3}\right) \times 20 = \frac{10000}{3} \text{ meters} \]

### Step 4: Determine the angle of sight with the vertical
The angle \( \theta \) can be found using the tangent function, where:

\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{R}{h} \]

Substituting the values:

\[ \tan(\theta) = \frac{\frac{10000}{3}}{1960} \]

Calculating this gives:

\[ \tan(\theta) = \frac{10000}{3 \times 1960} \approx \frac{10000}{5880} \approx 1.70 \]

### Step 5: Calculate the angle \( \theta \)
Now we can find \( \theta \) using the inverse tangent function:

\[ \theta = \tan^{-1}(1.70) \]

Using a calculator, we find:

\[ \theta \approx 60^\circ \]

### Conclusion
Thus, the pilot should release the survival kit at an angle of approximately **60 degrees** with respect to the vertical.

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