A balloon starts from the state of rest from the ground with constant acceleration g/n. after time T, a stone is dropped from the balloon. If stone takes time T to reach the ground then calculate value of n.

Explanation

To solve the problem step-by-step, we will analyze the motion of the balloon and the stone dropped from it.

### Step 1: Understanding the Motion of the Balloon
The balloon starts from rest and moves upward with a constant acceleration of \( \frac{g}{n} \). The initial velocity \( u \) is 0, and the time \( T \) is given.

Using the first equation of motion:
\[ v = u + at \]
where:
- \( v \) = final velocity of the balloon after time \( T \)
- \( u = 0 \) (initial velocity)
- \( a = \frac{g}{n} \) (acceleration)
- \( t = T \)

Substituting the values:
\[ v = 0 + \left(\frac{g}{n}\right) T = \frac{gT}{n} \]

### Step 2: Calculating the Height of the Balloon
Next, we calculate the height \( H \) attained by the balloon after time \( T \) using the second equation of motion:
\[ s = ut + \frac{1}{2} a t^2 \]
where:
- \( s = H \) (height)
- \( u = 0 \)
- \( a = \frac{g}{n} \)
- \( t = T \)

Substituting the values:
\[ H = 0 \cdot T + \frac{1}{2} \left(\frac{g}{n}\right) T^2 = \frac{gT^2}{2n} \]

### Step 3: Analyzing the Motion of the Stone
When the stone is dropped from the balloon, it has an initial upward velocity equal to the velocity of the balloon at that moment, which is \( \frac{gT}{n} \). The stone will then move under the influence of gravity, with an acceleration of \( -g \).

Using the second equation of motion for the stone:
\[ s = ut + \frac{1}{2} a t^2 \]
where:
- \( s = -H \) (the stone falls to the ground, so displacement is negative)
- \( u = \frac{gT}{n} \) (initial upward velocity of the stone)
- \( a = -g \) (acceleration due to gravity)
- \( t = T \)

Substituting the values:
\[ -H = \left(\frac{gT}{n}\right) T + \frac{1}{2} (-g) T^2 \]
\[ -H = \frac{gT^2}{n} - \frac{1}{2} g T^2 \]

### Step 4: Setting Up the Equation
We know from Step 2 that \( H = \frac{gT^2}{2n} \). Substituting this into the equation:
\[ -\frac{gT^2}{2n} = \frac{gT^2}{n} - \frac{1}{2} g T^2 \]

### Step 5: Simplifying the Equation
Multiplying through by \( -1 \):
\[ \frac{gT^2}{2n} = -\frac{gT^2}{n} + \frac{1}{2} g T^2 \]

Combining terms:
\[ \frac{gT^2}{2n} + \frac{gT^2}{n} = \frac{1}{2} g T^2 \]
\[ \frac{gT^2}{2n} + \frac{2gT^2}{2n} = \frac{1}{2} g T^2 \]
\[ \frac{3gT^2}{2n} = \frac{1}{2} g T^2 \]

### Step 6: Solving for \( n \)
Dividing both sides by \( gT^2 \) (assuming \( g \) and \( T^2 \) are not zero):
\[ \frac{3}{2n} = \frac{1}{2} \]
Cross-multiplying gives:
\[ 3 = n \]

### Conclusion
The value of \( n \) is \( 3 \).

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