An airplane with a speed of 97.5 m/s is climbing upward at an angle of `50.0^(@)` with respect to the horizontal. When the plane's altitude is 732 m, the pilot releases a package, (a) Calculate the distance along the ground, measured from a point directly beneath the point of release, to where the package hits the earth.

Options

A. `+425m`
B. `+1380m`
C. `-678m`
D. `-2880m`

Answer

To solve the problem of the package released from the airplane, we will follow these steps:

### Step 1: Determine the initial velocity components of the package.
The airplane is moving at a speed of 97.5 m/s at an angle of 50 degrees with respect to the horizontal. We need to find the horizontal and vertical components of this velocity.

- **Horizontal component (Vx)**:
\[ V_x = V \cdot \cos(\theta) = 97.5 \cdot \cos(50^\circ) \]

- **Vertical component (Vy)**:
\[ V_y = V \cdot \sin(\theta) = 97.5 \cdot \sin(50^\circ) \]

### Step 2: Calculate the vertical motion of the package.
The package is released from an altitude of 732 m. We will use the second equation of motion to find the time it takes for the package to hit the ground.

The equation is:
\[ h = V_y \cdot t + \frac{1}{2} a \cdot t^2 \]
Where:
- \( h = -732 \) m (the negative sign indicates a downward displacement),
- \( V_y \) is the initial vertical velocity calculated in Step 1,
- \( a = -9.8 \, \text{m/s}^2 \) (acceleration due to gravity).

### Step 3: Rearrange the equation into standard quadratic form.
Rearranging gives us:
\[ \frac{1}{2} a t^2 + V_y t + h = 0 \]
Substituting the values, we get:
\[ 4.9 t^2 - V_y t - 732 = 0 \]

### Step 4: Solve the quadratic equation for time (t).
Using the quadratic formula:
\[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Where:
- \( a = 4.9 \),
- \( b = -V_y \),
- \( c = -732 \).

### Step 5: Calculate the horizontal distance traveled.
Once we have the time \( t \) from Step 4, we can calculate the horizontal distance \( d \) traveled by the package using:
\[ d = V_x \cdot t \]

### Step 6: Final calculation and conclusion.
Substituting the values will give us the distance along the ground from the point directly beneath the release point to where the package hits the earth.

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