A balloon is going upwards with a constant velocity `15 m//s`. When the ballooon is at 50 m height, a stone is dropped outside from the balloon. How long will stone take to reach at the ground? `("take " g=10 m//s^(2))`

Explanation

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

### Step 1: Identify the initial conditions
- The balloon is moving upwards with a constant velocity of \( u = 15 \, \text{m/s} \).
- The height from which the stone is dropped is \( h = 50 \, \text{m} \).
- The acceleration due to gravity is \( g = 10 \, \text{m/s}^2 \) (acting downwards).

### Step 2: Determine the initial velocity of the stone
When the stone is dropped from the balloon, it retains the upward velocity of the balloon. Therefore, the initial velocity of the stone is:
- \( u = -15 \, \text{m/s} \) (negative because it is directed upwards, while we will consider downwards as positive).

### Step 3: Set up the kinematic equation
We can use the second equation of motion:
\[ s = ut + \frac{1}{2} a t^2 \]
Where:
- \( s \) is the displacement (which will be \( -50 \, \text{m} \) since it moves downwards),
- \( u = -15 \, \text{m/s} \),
- \( a = g = 10 \, \text{m/s}^2 \) (downwards).

Substituting these values into the equation gives:
\[ -50 = -15t + \frac{1}{2} \cdot 10 \cdot t^2 \]

### Step 4: Rearrange the equation
Rearranging the equation, we get:
\[ -50 = -15t + 5t^2 \]
\[ 5t^2 - 15t + 50 = 0 \]

### Step 5: Simplify the equation
To simplify, we can multiply through by -1:
\[ 5t^2 - 15t + 50 = 0 \]

### Step 6: Factor the quadratic equation
We can factor the quadratic equation:
\[ 5(t^2 - 3t - 10) = 0 \]
This can be factored as:
\[ (t - 5)(t + 2) = 0 \]

### Step 7: Solve for \( t \)
Setting each factor to zero gives us:
1. \( t - 5 = 0 \) → \( t = 5 \, \text{s} \)
2. \( t + 2 = 0 \) → \( t = -2 \, \text{s} \) (not valid since time cannot be negative)

Thus, the only valid solution is:
\[ t = 5 \, \text{s} \]

### Conclusion
The stone will take **5 seconds** to reach the ground.

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