A balloon from rest accelerates uniformly upward with 'a' `ms^(-2)`, for t seconds of time. A stone is released from the balloon. Now, read the following statements to pick the right ones.
(a) The stone's initial velocity is zero, relative to balloon
(b) The stone's initial velocity is nonzero, relative to earth
( c) The time taken to reach the ground from the balloon's frame of reference is inversely proportional to `sqrt((a+g))`
(d) The time take to reach the ground from earth's frame of reference is directly proportional to `sqrt((a+g))`

Options

A. a,b,c
B. a,c,d
C. a,b,d
D. a,c

Answer

To solve the problem, we need to analyze the motion of the balloon and the stone in detail. Let's break it down step by step.

### Step 1: Understanding the Motion of the Balloon
The balloon accelerates upward from rest with an acceleration 'a' for a time 't'. The initial velocity of the balloon (u) is 0 m/s, and it accelerates uniformly.

**Equation of Motion:**
The final velocity (v) of the balloon after time 't' can be calculated using the equation of motion:
\[ v = u + at \]
Since \( u = 0 \):
\[ v = 0 + at = at \]

### Step 2: Analyzing the Stone's Release
When the stone is released from the balloon, it has the same velocity as the balloon at that instant. Therefore, the initial velocity of the stone relative to the balloon is 0 m/s, but it has a non-zero velocity relative to the Earth.

**Initial Velocity of the Stone:**
- Relative to the balloon: \( v_{stone, balloon} = 0 \) m/s
- Relative to the Earth: \( v_{stone, earth} = at \) m/s

### Step 3: Considering Forces on the Stone
Once the stone is released, it is acted upon by gravity. The acceleration of the stone downward is \( g \) (acceleration due to gravity).

### Step 4: Time to Reach the Ground
To find the time taken for the stone to reach the ground from the balloon's frame of reference, we can use the following kinematic equation:
\[ s = ut + \frac{1}{2} a t^2 \]
Here, \( s \) is the distance to the ground, \( u \) is the initial velocity of the stone relative to the balloon (which is 0), and \( a \) is the effective acceleration (which is \( g + a \) since the balloon is moving upwards).

Thus, the equation simplifies to:
\[ s = \frac{1}{2} (g + a) t^2 \]

From this, we can derive that the time \( t \) taken to reach the ground is inversely proportional to \( \sqrt{(a + g)} \).

### Step 5: Time from Earth's Frame of Reference
In the Earth's frame of reference, the stone has an initial velocity of \( at \) and accelerates downwards with \( g \). The time taken to reach the ground can be derived from the same kinematic equations, leading to a relationship that shows the time taken is directly proportional to \( \sqrt{(a + g)} \).

### Conclusion
Now, let's summarize the correct statements based on our analysis:
- (a) The stone's initial velocity is zero, relative to the balloon. **(True)**
- (b) The stone's initial velocity is nonzero, relative to the earth. **(True)**
- (c) The time taken to reach the ground from the balloon's frame of reference is inversely proportional to \( \sqrt{(a + g)} \). **(True)**
- (d) The time taken to reach the ground from the earth's frame of reference is directly proportional to \( \sqrt{(a + g)} \). **(True)**

### Final Answers:
All statements (a), (b), (c), and (d) are correct.