A ball is thrown from the top of a tower with an intial velocity of 10 m//s at an angle `37^(@)` above the horizontal, hits the ground at a distance 16 m from the base of tower. Calculate height of tower. `[g=10 m//s^(2)]`

Explanation

To solve the problem step by step, we will break it down into manageable parts.

### Step 1: Understand the Problem
We have a ball thrown from the top of a tower with an initial velocity of 10 m/s at an angle of 37 degrees above the horizontal. It hits the ground at a distance of 16 m from the base of the tower. We need to find the height of the tower.

### Step 2: Resolve the Initial Velocity
The initial velocity can be resolved into horizontal and vertical components using trigonometric functions:
- \( u_x = u \cos(37^\circ) \)
- \( u_y = u \sin(37^\circ) \)

Given \( u = 10 \, \text{m/s} \):
- \( u_x = 10 \cos(37^\circ) \)
- \( u_y = 10 \sin(37^\circ) \)

Using the values of cosine and sine:
- \( \cos(37^\circ) = \frac{4}{5} \)
- \( \sin(37^\circ) = \frac{3}{5} \)

Calculating the components:
- \( u_x = 10 \times \frac{4}{5} = 8 \, \text{m/s} \)
- \( u_y = 10 \times \frac{3}{5} = 6 \, \text{m/s} \)

### Step 3: Calculate Time of Flight
Using the horizontal motion to find the time of flight:
- The horizontal distance covered is \( x = 16 \, \text{m} \).
- The formula for horizontal motion is:
\[ x = u_x \cdot t \]
Substituting the values:
\[ 16 = 8 \cdot t \]
Solving for \( t \):
\[ t = \frac{16}{8} = 2 \, \text{s} \]

### Step 4: Analyze Vertical Motion
Now, we will analyze the vertical motion to find the height of the tower. The vertical displacement \( h \) can be calculated using the kinematic equation:
\[ s = ut + \frac{1}{2} a t^2 \]
Here, \( s = -h \) (downward displacement), \( u = u_y = 6 \, \text{m/s} \), \( a = -g = -10 \, \text{m/s}^2 \), and \( t = 2 \, \text{s} \).

Substituting the values into the equation:
\[ -h = 6 \cdot 2 + \frac{1}{2} \cdot (-10) \cdot (2^2) \]
Calculating:
\[ -h = 12 - \frac{1}{2} \cdot 10 \cdot 4 \]
\[ -h = 12 - 20 \]
\[ -h = -8 \]
Thus, the height of the tower \( h \) is:
\[ h = 8 \, \text{m} \]

### Final Answer
The height of the tower is **8 meters**.

---