A balloon is at a height of 81 m and is ascending upwards with a velocity of 12 m/s . A body of 2kg weight is dropped from it. If `g = 10 m//s^(2)`, the body will reach the surface of the earth in

Options

A. 1.5s
B. 4.025s
C. 5.4 s
D. 7.75s

Answer

To solve the problem step by step, we can follow these steps:

### Step 1: Understand the problem
We have a balloon at a height of 81 m ascending with a velocity of 12 m/s. A body is dropped from the balloon, and we need to find out how long it takes for the body to reach the ground.

### Step 2: Identify the variables
- Initial height (H) = 81 m (upward)
- Initial velocity (U) = 12 m/s (upward)
- Acceleration due to gravity (g) = 10 m/s² (downward)
- Final height (h) = 0 m (ground level)

### Step 3: Set up the equation of motion
We can use the second equation of motion:
\[ H = U t + \frac{1}{2} a t^2 \]
Where:
- \( H \) is the displacement (final height - initial height)
- \( U \) is the initial velocity
- \( a \) is the acceleration (which will be negative because it is downward)
- \( t \) is the time taken.

Since the body is dropped from the balloon, we consider downward as negative. Therefore, we can rewrite the equation as:
\[ -81 = 12t - \frac{1}{2} (10)t^2 \]
This simplifies to:
\[ -81 = 12t - 5t^2 \]

### Step 4: Rearrange the equation
Rearranging gives us:
\[ 5t^2 - 12t - 81 = 0 \]

### Step 5: Solve the quadratic equation
We can use the quadratic formula:
\[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Where \( a = 5 \), \( b = -12 \), and \( c = -81 \).

Calculating the discriminant:
\[ b^2 - 4ac = (-12)^2 - 4 \cdot 5 \cdot (-81) \]
\[ = 144 + 1620 \]
\[ = 1764 \]

Now substituting back into the quadratic formula:
\[ t = \frac{12 \pm \sqrt{1764}}{2 \cdot 5} \]
\[ = \frac{12 \pm 42}{10} \]

### Step 6: Calculate the two possible values for t
Calculating the two possible values:
1. \( t = \frac{12 + 42}{10} = \frac{54}{10} = 5.4 \) seconds
2. \( t = \frac{12 - 42}{10} = \frac{-30}{10} = -3 \) seconds (not physically meaningful)

Thus, the time taken for the body to reach the ground is:
\[ t = 5.4 \text{ seconds} \]

### Final Answer
The body will reach the surface of the earth in **5.4 seconds**.

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