A body is dropped from a balloon meving up wigh a velocity of `4 m s^(-2) ` when the balloon is at a height of `12.5` m from the ground.
The distance of separation between of separation between the body and the balloon after `5` is.

Options

A. `122.5 m`
B. `100.5 m`
C. `132.5 m`
D. `112.5 m`

Explanation

To solve the problem, we need to determine the distance of separation between the body and the balloon after 5 seconds. Let's break down the solution step by step.

### Step 1: Understand the Initial Conditions
- The balloon is moving upward with a velocity of \( v_b = 4 \, \text{m/s} \).
- The height of the balloon when the body is dropped is \( h = 12.5 \, \text{m} \).

### Step 2: Determine the Motion of the Balloon
The balloon continues to move upward for 5 seconds. The distance traveled by the balloon in that time can be calculated using the formula:
\[ \text{Distance}_{\text{balloon}} = v_b \times t \]
Substituting the values:
\[ \text{Distance}_{\text{balloon}} = 4 \, \text{m/s} \times 5 \, \text{s} = 20 \, \text{m} \]

### Step 3: Calculate the Final Height of the Balloon
The final height of the balloon after 5 seconds can be calculated by adding the initial height to the distance traveled:
\[ \text{Final Height}_{\text{balloon}} = h + \text{Distance}_{\text{balloon}} = 12.5 \, \text{m} + 20 \, \text{m} = 32.5 \, \text{m} \]

### Step 4: Determine the Motion of the Body
When the body is dropped, it has an initial velocity equal to that of the balloon, which is \( v_0 = 4 \, \text{m/s} \) upward. The body will experience downward acceleration due to gravity, which is \( g = 9.8 \, \text{m/s}^2 \).

The position of the body after 5 seconds can be calculated using the kinematic equation:
\[ s = v_0 t + \frac{1}{2} a t^2 \]
Here, \( a = -g = -9.8 \, \text{m/s}^2 \) (since it is acting downward).
Substituting the values:
\[ s = 4 \, \text{m/s} \times 5 \, \text{s} + \frac{1}{2} \times (-9.8 \, \text{m/s}^2) \times (5 \, \text{s})^2 \]
Calculating the terms:
\[ s = 20 \, \text{m} - \frac{1}{2} \times 9.8 \times 25 \]
\[ s = 20 \, \text{m} - 122.5 \, \text{m} = -102.5 \, \text{m} \]
This negative value indicates that the body has fallen below the initial height of the balloon.

### Step 5: Calculate the Height of the Body Above the Ground
The height of the body above the ground after 5 seconds can be calculated as:
\[ \text{Height}_{\text{body}} = h + s = 12.5 \, \text{m} - 102.5 \, \text{m} = -90 \, \text{m} \]
Since height cannot be negative, we interpret this as the body being 90 m below the ground level.

### Step 6: Calculate the Distance of Separation
Finally, the distance of separation between the body and the balloon after 5 seconds is:
\[ \text{Distance of Separation} = \text{Final Height}_{\text{balloon}} - \text{Height}_{\text{body}} = 32.5 \, \text{m} - (-90 \, \text{m}) = 32.5 \, \text{m} + 90 \, \text{m} = 122.5 \, \text{m} \]

### Final Answer
The distance of separation between the body and the balloon after 5 seconds is **122.5 meters**.

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