A body is thrown horizontal from the top of a tower and strikes the ground after two seconds at angle of `45^(@)` with the horizontal. Find the height of the tower and the speed with which the body was thrown. Take `g = 9.8 ms^(-2)`.

Answer

To solve the problem, we need to find the height of the tower and the speed with which the body was thrown horizontally. We can break this down into steps:

### Step 1: Understand the motion of the body
The body is thrown horizontally from the top of the tower and strikes the ground after 2 seconds at an angle of 45 degrees with the horizontal. This means that the horizontal and vertical components of its velocity are equal when it strikes the ground.

### Step 2: Use the time of flight to find the height of the tower
The height (h) of the tower can be calculated using the second equation of motion for vertical displacement:
\[ h = ut + \frac{1}{2}gt^2 \]
Where:
- \(u\) is the initial vertical velocity (which is 0 since the body is thrown horizontally),
- \(g\) is the acceleration due to gravity (9.8 m/s²),
- \(t\) is the time of flight (2 seconds).

Substituting the values:
\[ h = 0 \cdot 2 + \frac{1}{2} \cdot 9.8 \cdot (2^2) \]
\[ h = 0 + \frac{1}{2} \cdot 9.8 \cdot 4 \]
\[ h = 0 + 19.6 = 19.6 \text{ m} \]

### Step 3: Find the vertical component of the velocity just before impact
Since the body strikes the ground at an angle of 45 degrees, the vertical component of the velocity (\(v_y\)) is equal to the horizontal component of the velocity (\(v_x\)). We can find \(v_y\) using the first equation of motion:
\[ v_y = u_y + gt \]
Where:
- \(u_y = 0\) (initial vertical velocity),
- \(g = 9.8 \text{ m/s}^2\),
- \(t = 2 \text{ s}\).

Substituting the values:
\[ v_y = 0 + 9.8 \cdot 2 \]
\[ v_y = 19.6 \text{ m/s} \]

### Step 4: Since the angle is 45 degrees, find the horizontal component of the velocity
At an angle of 45 degrees, the horizontal and vertical components of the velocity are equal:
\[ v_x = v_y = 19.6 \text{ m/s} \]

### Step 5: Conclusion
- The height of the tower is **19.6 meters**.
- The speed with which the body was thrown horizontally is **19.6 m/s**.