Height of Rising Balloon When Dropped Stone Hits Ground

To find the height of the balloon when the stone hits the ground, we first calculate the time taken by the stone to reach the ground.

Step 1: Motion of the stone When the stone is dropped from the balloon, it inherits the upward velocity of the balloon.

• Initial vertical position of the stone, $y_0 = 75\text{ m}$
• Initial velocity of the stone, $u = +10\text{ m/s}$ (upward)
• Acceleration of the stone, $a = -g = -10\text{ m/s}^2$ (downward)
• Final vertical position when it hits the ground, $y = 0\text{ m}$

The vertical displacement of the stone is $\Delta y = y - y_0 = -75\text{ m}$.

Using the second equation of motion: $\Delta y = ut + \frac{1}{2}at^2$

Substitute the given values: $-75 = 10t + \frac{1}{2}(-10)t^2$ $-75 = 10t - 5t^2$

Rearranging the equation into a standard quadratic form: $5t^2 - 10t - 75 = 0$

Dividing the equation by $5$: $t^2 - 2t - 15 = 0$

Factorizing the quadratic equation: $(t - 5)(t + 3) = 0$

Since time must be a non-negative quantity, we select: $t = 5\text{ s}$

Step 2: Motion of the balloon During this time interval of $t = 5\text{ s}$, the balloon continues to rise at a constant speed of $v = 10\text{ m/s}$.

The additional height gained by the balloon in $5\text{ s}$ is: $\Delta h = v \cdot t = 10\text{ m/s} \times 5\text{ s} = 50\text{ m}$

Step 3: Total height of the balloon The total height of the balloon above the ground when the stone hits the ground is: $H = h_0 + \Delta h = 75\text{ m} + 50\text{ m} = 125\text{ m}$

Thus, the correct option is D.