Two balls are projected simultaneously with the same speed from the top of a tower, one vertically upwards and the other vertically downwards. They reach the ground in 9 s and 4 s, respectively. The height of the tower is `(g=10 m//s^(2))`

Options

A. 90 m
B. 180 m
C. 270 m
D. 360 m

Explanation

To solve the problem, we need to find the height of the tower (H) from which two balls are projected simultaneously, one upwards and the other downwards. We know the time taken for each ball to reach the ground: the ball thrown upwards takes 9 seconds, and the ball thrown downwards takes 4 seconds. We will use the equations of motion to derive the height of the tower.

### Step-by-Step Solution:

1. **Define Variables:**
- Let the initial speed of both balls be \( U \).
- Let the height of the tower be \( H \).
- The acceleration due to gravity \( g = 10 \, \text{m/s}^2 \).

2. **Equation for the Ball Thrown Downwards:**
- The ball thrown downwards takes 4 seconds to reach the ground.
- Using the equation of motion:
\[ H = Ut + \frac{1}{2} g t^2 \]
- For the downward ball:
\[ H = U(4) + \frac{1}{2} (10)(4^2) \]
\[ H = 4U + \frac{1}{2} \times 10 \times 16 \]
\[ H = 4U + 80 \]
- This is our **Equation 1**:
\[ H = 4U + 80 \quad \text{(1)} \]

3. **Equation for the Ball Thrown Upwards:**
- The ball thrown upwards takes 9 seconds to reach the ground.
- Using the same equation of motion:
\[ H = Ut - \frac{1}{2} g t^2 \]
- For the upward ball:
\[ H = U(9) - \frac{1}{2} (10)(9^2) \]
\[ H = 9U - \frac{1}{2} \times 10 \times 81 \]
\[ H = 9U - 405 \]
- This is our **Equation 2**:
\[ H = 9U - 405 \quad \text{(2)} \]

4. **Equate the Two Equations:**
- From Equation 1 and Equation 2, we can set them equal to each other:
\[ 4U + 80 = 9U - 405 \]

5. **Solve for U:**
- Rearranging gives:
\[ 405 + 80 = 9U - 4U \]
\[ 485 = 5U \]
\[ U = \frac{485}{5} = 97 \, \text{m/s} \]

6. **Substitute U Back to Find H:**
- Now substitute \( U \) back into either Equation 1 or Equation 2 to find \( H \). We'll use Equation 1:
\[ H = 4(97) + 80 \]
\[ H = 388 + 80 \]
\[ H = 468 \, \text{m} \]

### Final Answer:
The height of the tower is \( H = 468 \, \text{m} \).