A balloon rises with uniform velocity of `10 ms^(-1)`. When the balloon is at a height of 75 m if a stone is dropped from it the time taken by it to reach the ground is

Options

A. 3 sec
B. 4 sec
C. 5 sec
D. 7 sec

Answer

To solve the problem of how long it takes for the stone to reach the ground after being dropped from the balloon, we can follow these steps:

### Step 1: Understand the scenario
The balloon is rising with a uniform velocity of 10 m/s. When the balloon is at a height of 75 m, a stone is dropped from it. We need to find the time it takes for the stone to hit the ground.

### Step 2: Define the variables
- Initial velocity of the stone, \( u = -10 \, \text{m/s} \) (upward direction is negative)
- Height from which the stone is dropped, \( s = 75 \, \text{m} \) (downward direction is positive)
- Acceleration due to gravity, \( a = 10 \, \text{m/s}^2 \) (downward direction is positive)

### Step 3: Use the kinematic equation
We will use the equation of motion:
\[ s = ut + \frac{1}{2} a t^2 \]
Substituting the known values:
\[ 75 = -10t + \frac{1}{2} \times 10 \times t^2 \]
This simplifies to:
\[ 75 = -10t + 5t^2 \]

### Step 4: Rearrange the equation
Rearranging gives us:
\[ 5t^2 - 10t - 75 = 0 \]
Dividing the entire equation by 5:
\[ t^2 - 2t - 15 = 0 \]

### Step 5: Factor the quadratic equation
To solve for \( t \), we can factor the quadratic:
\[ (t - 5)(t + 3) = 0 \]
Setting each factor to zero gives:
\[ t - 5 = 0 \quad \text{or} \quad t + 3 = 0 \]
Thus, we find:
\[ t = 5 \quad \text{or} \quad t = -3 \]

### Step 6: Determine the valid solution
Since time cannot be negative, we discard \( t = -3 \) and accept:
\[ t = 5 \, \text{seconds} \]

### Conclusion
The time taken by the stone to reach the ground after being dropped from the balloon is **5 seconds**.

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