A balloon is ascending at the rate of `9.8 m//s` and is 39.2 m above the ground when a package is dropped. (a) How long does the package take to reach the ground? (b) with what speed does it hit the ground ? `(g = 9.8 m//s^(2))`

Explanation

To solve the problem step by step, we will break it down into two parts: (a) calculating the time it takes for the package to reach the ground, and (b) determining the speed at which it hits the ground.

### Given Data:
- Initial height of the package (h) = 39.2 m
- Initial velocity of the package (u) = 9.8 m/s (upward, since the balloon is ascending)
- Acceleration due to gravity (g) = 9.8 m/s² (downward)

### (a) How long does the package take to reach the ground?

1. **Identify the motion 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 -39.2 m, as it moves downward),
- \(u\) is the initial velocity (9.8 m/s),
- \(a\) is the acceleration (which will be -9.8 m/s², since it acts downward).

2. **Substituting the values**:
\[ -39.2 = 9.8t - \frac{1}{2} (9.8)t^2 \]
Rearranging gives:
\[ 0 = -4.9t^2 + 9.8t + 39.2 \]

3. **Rearranging into standard quadratic form**:
\[ 4.9t^2 - 9.8t - 39.2 = 0 \]

4. **Using the quadratic formula**:
The quadratic formula is given by:
\[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \(a = 4.9\), \(b = -9.8\), and \(c = -39.2\).

5. **Calculating the discriminant**:
\[ b^2 - 4ac = (-9.8)^2 - 4 \cdot 4.9 \cdot (-39.2) \]
\[ = 96.04 + 768.16 = 864.2 \]

6. **Finding the roots**:
\[ t = \frac{9.8 \pm \sqrt{864.2}}{2 \cdot 4.9} \]
\[ t = \frac{9.8 \pm 29.4}{9.8} \]
This gives two potential solutions for \(t\):
\[ t_1 = \frac{39.2}{9.8} = 4 \text{ seconds (valid)} \]
\[ t_2 = \frac{-19.6}{9.8} = -2 \text{ seconds (not valid)} \]

Thus, the time taken for the package to reach the ground is **4 seconds**.

### (b) With what speed does it hit the ground?

1. **Using the first equation of motion**:
The final velocity \(v\) can be calculated using:
\[ v = u + at \]
where \(u = 9.8 \text{ m/s}\), \(a = -9.8 \text{ m/s}^2\), and \(t = 4 \text{ s}\).

2. **Substituting the values**:
\[ v = 9.8 + (-9.8 \cdot 4) \]
\[ v = 9.8 - 39.2 \]
\[ v = -29.4 \text{ m/s} \]

The negative sign indicates that the direction of the velocity is downward. Thus, the speed at which it hits the ground is **29.4 m/s**.

### Summary of the Answers:
(a) Time taken to reach the ground = **4 seconds**
(b) Speed at which it hits the ground = **29.4 m/s**