A man in a balloon rising vertically with an accelration fo `4.9 ms^(-2)` released a ball `2 seconds` after the balloon is let fo from the fround. The greatst height above the ground reached by the ball is .

Options

A. (a) ` 9.8 m`
B. (b) ` 14.7 m`
C. (c ) ` 19 . 6 m`
D. (d) ` 24 . 5 m`

Explanation

To solve the problem step by step, we will first analyze the motion of the balloon and then the motion of the ball after it is released.

### Step 1: Determine the velocity of the balloon when the ball is released
- The balloon starts from rest, so its initial velocity \( u = 0 \).
- The acceleration of the balloon is given as \( a = 4.9 \, \text{m/s}^2 \).
- The time \( t \) until the ball is released is \( 2 \, \text{s} \).

Using the equation of motion:
\[ v = u + at \]
Substituting the values:
\[ v = 0 + (4.9 \, \text{m/s}^2)(2 \, \text{s}) = 9.8 \, \text{m/s} \]
So, the velocity of the balloon at the moment the ball is released is \( 9.8 \, \text{m/s} \) upwards.

### Step 2: Calculate the height of the balloon at the time the ball is released
Using the equation for distance traveled under constant acceleration:
\[ s = ut + \frac{1}{2} a t^2 \]
Substituting the known values:
\[ s = 0 + \frac{1}{2} (4.9 \, \text{m/s}^2)(2 \, \text{s})^2 = \frac{1}{2} (4.9)(4) = 9.8 \, \text{m} \]
Thus, the height of the balloon when the ball is released is \( 9.8 \, \text{m} \).

### Step 3: Determine the maximum height the ball reaches after being released
When the ball is released, it has an initial upward velocity of \( 9.8 \, \text{m/s} \) and is subject to gravitational acceleration downwards, which is \( -9.8 \, \text{m/s}^2 \).

Using the equation:
\[ v^2 = u^2 + 2as \]
Where:
- Final velocity \( v = 0 \) (at the highest point),
- Initial velocity \( u = 9.8 \, \text{m/s} \),
- Acceleration \( a = -9.8 \, \text{m/s}^2 \).

Rearranging the equation to solve for \( s \):
\[ 0 = (9.8)^2 + 2(-9.8)s \]
\[ 0 = 96.04 - 19.6s \]
\[ 19.6s = 96.04 \]
\[ s = \frac{96.04}{19.6} = 4.9 \, \text{m} \]
So, the ball rises an additional \( 4.9 \, \text{m} \) above the point of release.

### Step 4: Calculate the total height above the ground
The total height \( H \) above the ground is the height of the balloon when the ball is released plus the additional height the ball rises:
\[ H = 9.8 \, \text{m} + 4.9 \, \text{m} = 14.7 \, \text{m} \]

### Final Answer
The greatest height above the ground reached by the ball is \( 14.7 \, \text{m} \).

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