A balloon moves up vertically such that if a stone is thrown from it with a horizontal velocity `v_(0)` relative to it the stone always hits the ground at a fixed point `2v_(0)^(2)//g` horizontally away from it. Find the height of the balloon as a function of time.

Answer

To solve the problem of finding the height of the balloon as a function of time, we can follow these steps:

### Step 1: Understand the problem
The balloon moves upwards with a vertical speed \( v_0 \). When a stone is thrown horizontally from the balloon with the same speed \( v_0 \), it lands at a horizontal distance of \( \frac{2v_0^2}{g} \) from the balloon. We need to find the height of the balloon as a function of time \( t \).

### Step 2: Determine the time of flight
The horizontal distance \( R \) covered by the stone is given by the formula:
\[ R = v_0 \cdot t \]
where \( t \) is the time of flight. Given that \( R = \frac{2v_0^2}{g} \), we can set up the equation:
\[ \frac{2v_0^2}{g} = v_0 \cdot t \]
Dividing both sides by \( v_0 \) (assuming \( v_0 \neq 0 \)):
\[ t = \frac{2v_0}{g} \tag{1} \]

### Step 3: Relate height to time
The height \( h \) of the balloon can be expressed using the equation of motion:
\[ h = ut + \frac{1}{2} a t^2 \]
Here, \( u = v_0 \) (the initial vertical speed of the balloon), \( a = g \) (the acceleration due to gravity, acting downwards), and \( t \) is the time of flight. Therefore, we can write:
\[ h = v_0 t - \frac{1}{2} g t^2 \]
Substituting \( t \) from equation (1):
\[ h = v_0 \left(\frac{2v_0}{g}\right) - \frac{1}{2} g \left(\frac{2v_0}{g}\right)^2 \]

### Step 4: Simplify the height equation
Now, substituting \( t \) into the height equation:
\[ h = \frac{2v_0^2}{g} - \frac{1}{2} g \cdot \frac{4v_0^2}{g^2} \]
This simplifies to:
\[ h = \frac{2v_0^2}{g} - \frac{2v_0^2}{g} = 0 \]
This indicates that the height is not constant, so we need to express \( h \) as a function of time.

### Step 5: Express height as a function of time
Using the time of flight \( t \):
\[ h(t) = v_0 t - \frac{1}{2} g t^2 \]
This equation gives us the height of the balloon as a function of time \( t \).

### Final Result
Thus, the height of the balloon as a function of time \( t \) is:
\[ h(t) = v_0 t - \frac{1}{2} g t^2 \]