A ball projected upwards from the foot of a tower. The ball crosses the top of the tower twice after an interval of `6 s` and the ball reaches the ground after `12 s`. The height of the tower is `(g = 10 m//s^2)` :

Options

A. 120 m
B. 135 m
C. 175 m
D. 80 m

Explanation

To solve the problem step by step, we need to analyze the motion of the ball projected upwards from the foot of a tower.

### Step 1: Understand the Motion
The ball is projected upwards and crosses the top of the tower twice. The time taken to reach the ground is 12 seconds, and it crosses the top of the tower after an interval of 6 seconds. This indicates that the ball reaches its maximum height at 6 seconds.

**Hint:** The total time of flight (12 seconds) can be divided into two equal halves: the ascent (6 seconds) and the descent (6 seconds).

### Step 2: Determine the Time to Maximum Height
Since the ball takes 12 seconds to reach the ground, it takes 6 seconds to reach the maximum height. The time taken to cross the top of the tower twice is 6 seconds, which means the ball reaches the top of the tower at 3 seconds after being projected upwards and again at 3 seconds before it hits the ground.

**Hint:** The time to reach maximum height is equal to half of the total time of flight.

### Step 3: Use the Second Equation of Motion
Using the second equation of motion:
\[ S = ut + \frac{1}{2}gt^2 \]
where:
- \( S \) is the displacement,
- \( u \) is the initial velocity,
- \( g \) is the acceleration due to gravity (10 m/s²),
- \( t \) is the time.

For the downward motion from the maximum height to the ground (6 seconds):
\[ S_1 = 0 \cdot 6 + \frac{1}{2} \cdot 10 \cdot (6^2) = 0 + 180 = 180 \, \text{m} \]

**Hint:** The initial velocity at the maximum height is zero because the ball stops rising before descending.

### Step 4: Calculate the Height of the Tower
For the upward motion from the foot of the tower to the top of the tower (3 seconds):
\[ S_2 = u \cdot 3 + \frac{1}{2} \cdot (-10) \cdot (3^2) \]
Let \( u \) be the initial velocity. We need to express \( u \) in terms of the distance traveled.

From the maximum height to the ground:
\[ S_1 = 180 \, \text{m} \]
From the foot of the tower to the top:
\[ S_2 = u \cdot 3 - \frac{1}{2} \cdot 10 \cdot 9 \]
\[ S_2 = 3u - 45 \]

### Step 5: Relate the Two Distances
The height of the tower \( h \) is given by:
\[ h = S_1 - S_2 \]
Substituting the values:
\[ h = 180 - (3u - 45) \]
\[ h = 180 - 3u + 45 \]
\[ h = 225 - 3u \]

### Step 6: Find the Initial Velocity
To find \( u \), we can use the time of flight to maximum height:
Using the first half of the motion (upwards):
\[ 0 = u - gt \]
At maximum height (6 seconds):
\[ 0 = u - 10 \cdot 6 \]
\[ u = 60 \, \text{m/s} \]

### Step 7: Calculate the Height of the Tower
Substituting \( u \) back into the height equation:
\[ h = 225 - 3(60) \]
\[ h = 225 - 180 \]
\[ h = 45 \, \text{m} \]

### Final Answer
The height of the tower is **45 meters**.