A ball is dropped from a balloon going up at a speed of 7 m/s. If the balloon was at a height 60 m at the time of dropping the ball, how long will the ball take in reaching the ground?

Explanation

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

### Step 1: Identify the given values
- Initial height of the balloon (h) = 60 m
- Initial velocity of the ball (u) = 7 m/s (upward)
- Acceleration due to gravity (g) = 9.8 m/s² (downward)

### Step 2: Set up the equation of motion
We will use the second equation of motion:
\[ s = ut + \frac{1}{2} a t^2 \]
Where:
- \( s \) = displacement (which will be -60 m since the ball is falling down)
- \( u \) = initial velocity (7 m/s, but since it's upward, we consider it as positive)
- \( a \) = acceleration (which is -g = -9.8 m/s²)

Substituting the values into the equation:
\[ -60 = 7t - \frac{1}{2} (9.8) t^2 \]

### Step 3: Rearrange the equation
Rearranging gives:
\[ -60 = 7t - 4.9t^2 \]
This can be rewritten as:
\[ 4.9t^2 - 7t - 60 = 0 \]

### Step 4: Solve the quadratic equation
We can use the quadratic formula:
\[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Where:
- \( a = 4.9 \)
- \( b = -7 \)
- \( c = -60 \)

Calculating the discriminant:
\[ b^2 - 4ac = (-7)^2 - 4(4.9)(-60) \]
\[ = 49 + 1176 = 1225 \]

Now substituting into the quadratic formula:
\[ t = \frac{-(-7) \pm \sqrt{1225}}{2(4.9)} \]
\[ = \frac{7 \pm 35}{9.8} \]

Calculating the two possible values for \( t \):
1. \( t = \frac{42}{9.8} \approx 4.2857 \) seconds (valid)
2. \( t = \frac{-28}{9.8} \) (not valid since time cannot be negative)

### Step 5: Final answer
The time taken by the ball to reach the ground is approximately:
\[ t \approx 4.29 \text{ seconds} \]

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