A stone thrown upward with a speed u from the top of a tower reaches the ground with a velocity 4u. The height of the tower is

Options

A. `(15u^(2))/(2g)`
B. `(7u^(2))/(2g)`
C. `(16u^(2))/(g)`
D. Zero

Answer

To solve the problem, we need to analyze the motion of the stone thrown upward from the top of the tower and its subsequent fall to the ground. Here’s a step-by-step solution:

### Step 1: Understand the motion of the stone
The stone is thrown upward with an initial velocity \( u \) and reaches a maximum height before falling back down to the ground. When it reaches the ground, its final velocity is \( 4u \).

### Step 2: Use the kinematic equation
We can use the kinematic equation for motion under uniform acceleration:
\[ v^2 = u^2 + 2a s \]
where:
- \( v \) is the final velocity (when the stone reaches the ground, \( v = 4u \)),
- \( u \) is the initial velocity (when the stone is thrown, \( u = u \)),
- \( a \) is the acceleration (which is \( g \) downward, so \( a = -g \)),
- \( s \) is the displacement (which is the height of the tower \( h \)).

### Step 3: Substitute values into the equation
Substituting the known values into the equation:
\[ (4u)^2 = u^2 + 2(-g)(-h) \]
This simplifies to:
\[ 16u^2 = u^2 + 2gh \]

### Step 4: Rearrange the equation
Rearranging the equation gives:
\[ 16u^2 - u^2 = 2gh \]
\[ 15u^2 = 2gh \]

### Step 5: Solve for height \( h \)
Now, we can solve for \( h \):
\[ h = \frac{15u^2}{2g} \]

### Conclusion
The height of the tower is:
\[ h = \frac{15u^2}{2g} \]

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