Question

A gas balloon is going up with a constant velocity of 10 m/s. When this balloon reached a height of 75 m, a stone is dropped from it and balloon keeps moving up with the same velocity. The height of the balloon when the stone hits the ground is ________ m. (Take \(g = 10\) m/s²)

• A. 85
• B. 150
• C. 129
• D. 125

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
When an object is dropped from a moving body, it inherits the instantaneous velocity of that body. Thus, the stone starts with an initial upward velocity. We calculate the time it takes for the stone to hit the ground. During this time, the balloon continues to travel upwards at its constant velocity.
Step 2: Key Formula or Approach:
1. Equation of motion for the stone: $S = ut + \frac{1}{2}at^2$.
2. Here, $u = +10\text{ m/s}$, $a = -g = -10\text{ m/s}^2$, and displacement $S = -75\text{ m}$ (since it hits the ground below the release point).
3. Final height of the balloon: $H_{final} = H_{initial} + v_{balloon} \times t$.
Step 3: Detailed Explanation:
For the dropped stone:
Initial velocity $u = 10\text{ m/s}$ (upwards, inherited from the balloon)
Displacement $S = -75\text{ m}$ (downwards to the ground)
Acceleration $a = -10\text{ m/s}^2$
Using the second equation of motion:
$S = ut + \frac{1}{2}at^2$
$-75 = 10t + \frac{1}{2}(-10)t^2$
$-75 = 10t - 5t^2$
Divide the entire equation by $-5$:
$15 = -2t + t^2$
$t^2 - 2t - 15 = 0$.
Factor the quadratic equation:
$(t - 5)(t + 3) = 0$.
Since time cannot be negative, $t = 5\text{ s}$.
So, it takes $5\text{ seconds}$ for the stone to hit the ground.
Meanwhile, the balloon has continued to ascend at a constant velocity of $10\text{ m/s}$ for these $5\text{ seconds}$.
Distance traveled by the balloon in this time $= v \times t = 10 \times 5 = 50\text{ m}$.
Total height of the balloon when the stone hits the ground:
$H_{total} = H_{initial} + \text{Distance moved}$
$H_{total} = 75\text{ m} + 50\text{ m} = 125\text{ m}$.
Step 4: Final Answer:
The height of the balloon is $125\text{ m}$.