A packet is dropped from a balloon rising up with uniform velocity `9.8 "ms"^(-1)`. If the balloon is at a height of 39.2 m from the ground at the time of dropping the stone, the stone reaches the ground after

Options

A. 2 s
B. 6s
C. 4 s
D. 8s

Answer

To solve the problem, we need to determine the time it takes for the packet (stone) to reach the ground after being dropped from a balloon that is rising with a uniform velocity. Here’s a step-by-step solution:

### Step 1: Identify the given data
- Initial velocity of the stone (u) = 9.8 m/s (upward, same as the balloon)
- Height from which the stone is dropped (s) = -39.2 m (negative because it is falling down)
- Acceleration due to gravity (a) = -9.8 m/s² (negative because it acts downward)

### Step 2: Use the second equation of motion
The second equation of motion is given by:
\[ s = ut + \frac{1}{2} a t^2 \]

Substituting the known values into the equation:
\[ -39.2 = 9.8t + \frac{1}{2}(-9.8)t^2 \]

### Step 3: Simplify the equation
This simplifies to:
\[ -39.2 = 9.8t - 4.9t^2 \]

Rearranging gives:
\[ 4.9t^2 - 9.8t - 39.2 = 0 \]

### Step 4: Multiply through by 10 to eliminate decimals
To make calculations easier, multiply the entire equation by 10:
\[ 49t^2 - 98t - 392 = 0 \]

### Step 5: Divide the equation by 7
To simplify further, divide the entire equation by 7:
\[ 7t^2 - 14t - 56 = 0 \]

### Step 6: Factor the quadratic equation
We can factor the quadratic equation:
\[ (t - 8)(t + 7) = 0 \]

### Step 7: Solve for t
Setting each factor to zero gives:
1. \( t - 8 = 0 \) → \( t = 8 \)
2. \( t + 7 = 0 \) → \( t = -7 \) (not valid since time cannot be negative)

Thus, the valid solution is:
\[ t = 8 \text{ seconds} \]

### Final Answer
The stone reaches the ground after **8 seconds**.

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