A balloon starts rising from ground from rest with an upward acceleration `2m//s^(2)`. Just after 1 s, a stone is dropped from it. The time taken by stone to strike the ground is nearly

Options

A. 0.3 s
B. 0.7 s
C. 1 s
D. 1.4 s

Answer

To solve the problem step by step, we will follow the outlined approach in the video transcript.

### Step 1: Determine the height of the balloon after 1 second
The balloon starts from rest and rises with an upward acceleration of \(2 \, \text{m/s}^2\). We can use the second equation of motion to find the distance covered by the balloon in 1 second.

The formula is:
\[ s = ut + \frac{1}{2} a t^2 \]
Where:
- \(u = 0 \, \text{m/s}\) (initial velocity)
- \(a = 2 \, \text{m/s}^2\) (acceleration)
- \(t = 1 \, \text{s}\)

Substituting the values:
\[ s = 0 \cdot 1 + \frac{1}{2} \cdot 2 \cdot (1)^2 = 0 + 1 = 1 \, \text{m} \]

### Step 2: Determine the velocity of the balloon at 1 second
Next, we need to find the velocity of the balloon at the moment the stone is dropped. We can use the first equation of motion:
\[ v = u + at \]
Where:
- \(u = 0 \, \text{m/s}\) (initial velocity)
- \(a = 2 \, \text{m/s}^2\) (acceleration)
- \(t = 1 \, \text{s}\)

Substituting the values:
\[ v = 0 + 2 \cdot 1 = 2 \, \text{m/s} \]

### Step 3: Analyze the motion of the stone after it is dropped
When the stone is dropped from the balloon, it has an initial upward velocity of \(2 \, \text{m/s}\) and it will experience downward acceleration due to gravity, which we take as \(g = 10 \, \text{m/s}^2\).

We need to find the time \(t\) it takes for the stone to hit the ground. The stone will move upwards for a certain time before it starts descending.

Using the second equation of motion for the stone:
\[ s = ut + \frac{1}{2} (-g) t^2 \]
Where:
- \(s = -1 \, \text{m}\) (the stone falls 1 meter to the ground)
- \(u = 2 \, \text{m/s}\) (initial upward velocity)
- \(g = 10 \, \text{m/s}^2\) (downward acceleration)

Substituting the values:
\[ -1 = 2t - \frac{1}{2} \cdot 10 t^2 \]
This simplifies to:
\[ -1 = 2t - 5t^2 \]
Rearranging gives:
\[ 5t^2 - 2t - 1 = 0 \]

### Step 4: Solve the quadratic equation
Now, we can solve this quadratic equation using the quadratic formula:
\[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Where:
- \(a = 5\)
- \(b = -2\)
- \(c = -1\)

Substituting the values:
\[ t = \frac{-(-2) \pm \sqrt{(-2)^2 - 4 \cdot 5 \cdot (-1)}}{2 \cdot 5} \]
\[ t = \frac{2 \pm \sqrt{4 + 20}}{10} \]
\[ t = \frac{2 \pm \sqrt{24}}{10} \]
\[ t = \frac{2 \pm 2\sqrt{6}}{10} \]
\[ t = \frac{1 \pm \sqrt{6}}{5} \]

### Step 5: Calculate the time
We take the positive root since time cannot be negative:
\[ t \approx \frac{1 + 2.45}{5} \approx \frac{3.45}{5} \approx 0.69 \, \text{s} \]

Thus, the time taken by the stone to strike the ground is approximately \(0.7 \, \text{s}\).