A level flight olane flying at an altitude of 1024 ft with a speed of 240 ft/s is overtaking a motor boat travelling at 80 ft/s in the same direction as the plane. At what horizontal distance before the boat should a bag be dropped from the plane in order to hit the boat ? ` [ g = 32 " ft/s"^(2)` ]

Options

A. 1000 ft
B. 1280 ft
C. 980 ft
D. 1200 ft

Answer

To solve the problem of determining the horizontal distance before the boat from which a bag should be dropped from a plane in order to hit the boat, we can follow these steps:

### Step 1: Identify the given data
- Altitude of the plane (h) = 1024 ft
- Speed of the plane (V_plane) = 240 ft/s
- Speed of the boat (V_boat) = 80 ft/s
- Acceleration due to gravity (g) = 32 ft/s²

### Step 2: Calculate the relative speed of the plane with respect to the boat
Since both the plane and the boat are moving in the same direction, the relative speed (V_relative) is given by:
\[ V_{relative} = V_{plane} - V_{boat} \]
Substituting the values:
\[ V_{relative} = 240 \, \text{ft/s} - 80 \, \text{ft/s} = 160 \, \text{ft/s} \]

### Step 3: Calculate the time taken for the bag to fall to the water
Using the equation of motion for vertical displacement:
\[ s = ut + \frac{1}{2} a t^2 \]
Where:
- \( s \) = vertical displacement = -h (since it is falling down)
- \( u \) = initial vertical velocity = 0 (the bag is dropped)
- \( a \) = acceleration = -g (downward)

Substituting the values:
\[ -h = 0 \cdot t + \frac{1}{2} (-g) t^2 \]
This simplifies to:
\[ -h = -\frac{1}{2} g t^2 \]
Rearranging gives:
\[ h = \frac{1}{2} g t^2 \]
Now substituting \( g = 32 \, \text{ft/s}^2 \):
\[ 1024 = \frac{1}{2} \cdot 32 \cdot t^2 \]
\[ 1024 = 16 t^2 \]
\[ t^2 = \frac{1024}{16} = 64 \]
Taking the square root:
\[ t = \sqrt{64} = 8 \, \text{s} \]

### Step 4: Calculate the horizontal distance the bag travels
The horizontal distance (d) can be calculated using the formula:
\[ d = V_{relative} \cdot t \]
Substituting the values:
\[ d = 160 \, \text{ft/s} \cdot 8 \, \text{s} \]
\[ d = 1280 \, \text{ft} \]

### Conclusion
The horizontal distance before the boat from which the bag should be dropped is **1280 feet**.

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