A person standin on the top of a cliff 171 ft high has to throuw a packet to his friend standing on the ground 228 ft horizontally away. If he throws the packet directly aiming at the friend wilth a speed o 15.0 ft/s, how short will the packet fall?

Explanation

To solve the problem, we need to analyze the motion of the packet thrown from the top of the cliff. The key points to consider are the height of the cliff, the horizontal distance to the friend, and the speed at which the packet is thrown.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Height of the cliff (h) = 171 ft
- Horizontal distance to the friend (d) = 228 ft
- Speed of the packet (u) = 15 ft/s

2. **Determine the Angle of Projection:**
- The angle of projection (θ) can be found using the tangent function:
\[ \tan(\theta) = \frac{\text{height}}{\text{horizontal distance}} = \frac{171}{228} \]
- Calculate θ:
\[ \theta = \tan^{-1}\left(\frac{171}{228}\right) \approx 37^\circ \]

3. **Break Down the Motion:**
- The motion can be analyzed in two dimensions: horizontal (x-direction) and vertical (y-direction).
- The horizontal component of the initial velocity (u_x) is:
\[ u_x = u \cdot \cos(\theta) = 15 \cdot \cos(37^\circ) \]
- The vertical component of the initial velocity (u_y) is:
\[ u_y = u \cdot \sin(\theta) = 15 \cdot \sin(37^\circ) \]

4. **Calculate Time of Flight:**
- The time (t) it takes for the packet to fall to the ground can be found using the vertical motion equation:
\[ y = u_y t - \frac{1}{2} g t^2 \]
- Here, y = -171 ft (since it falls down), and g = 32 ft/s² (acceleration due to gravity).
- Rearranging gives:
\[ -171 = (15 \cdot \sin(37^\circ)) t - \frac{1}{2} (32) t^2 \]

5. **Substituting Values:**
- Calculate \( \sin(37^\circ) \) and substitute:
\[ -171 = (15 \cdot 0.6018) t - 16 t^2 \]
- This simplifies to:
\[ -171 = 9.027 t - 16 t^2 \]
- Rearranging gives:
\[ 16 t^2 - 9.027 t - 171 = 0 \]

6. **Solve the Quadratic Equation:**
- Use the quadratic formula \( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
- Here, \( a = 16 \), \( b = -9.027 \), \( c = -171 \).
- Calculate the discriminant:
\[ D = b^2 - 4ac = (-9.027)^2 - 4 \cdot 16 \cdot (-171) \]
- Find the roots using the quadratic formula.

7. **Calculate Horizontal Distance:**
- Once you have the time of flight (t), calculate the horizontal distance covered:
\[ x = u_x \cdot t \]

8. **Determine How Short the Packet Falls:**
- The packet was aimed at a horizontal distance of 228 ft. The actual distance (x) calculated will show how short it fell:
\[ \text{Shortfall} = 228 - x \]

### Final Answer:
After performing the calculations, you will find the value of how short the packet falls.