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 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)
Updated On: Apr 13, 2026
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The Correct Option is D
Solution and Explanation
Step 1: Understanding the Concept:
When a stone is dropped from a moving balloon, it inherits the balloon's velocity at that instant due to inertia. We need to find the time the stone takes to hit the ground and then calculate the balloon's additional displacement during that same time interval.
Step 2: Key Formula or Approach:
1. Equation of motion for the stone: \(s = ut + \frac{1}{2}at^2\).
2. Total height of balloon: \(H_{final} = H_{initial} + (v_{balloon} \times t)\).
Step 3: Detailed Explanation:
For the stone: Initial velocity \(u = +10\) m/s (upward), displacement \(s = -75\) m (downward), \(a = -g = -10\) m/s². \[ -75 = 10t - \frac{1}{2}(10)t^2 \] \[ -75 = 10t - 5t^2 \implies 5t^2 - 10t - 75 = 0 \] \[ t^2 - 2t - 15 = 0 \implies (t - 5)(t + 3) = 0 \] Since time cannot be negative, \(t = 5\) s. During these 5 seconds, the balloon continues to move up at 10 m/s: \[ \Delta H_{balloon} = 10 \times 5 = 50 \text{ m} \] Total height of the balloon when the stone hits the ground: \[ H = 75 + 50 = 125 \text{ m} \]
Step 4: Final Answer:
The height of the balloon is 125 m.
When a stone is dropped from a moving balloon, it inherits the balloon's velocity at that instant due to inertia. We need to find the time the stone takes to hit the ground and then calculate the balloon's additional displacement during that same time interval.
Step 2: Key Formula or Approach:
1. Equation of motion for the stone: \(s = ut + \frac{1}{2}at^2\).
2. Total height of balloon: \(H_{final} = H_{initial} + (v_{balloon} \times t)\).
Step 3: Detailed Explanation:
For the stone: Initial velocity \(u = +10\) m/s (upward), displacement \(s = -75\) m (downward), \(a = -g = -10\) m/s². \[ -75 = 10t - \frac{1}{2}(10)t^2 \] \[ -75 = 10t - 5t^2 \implies 5t^2 - 10t - 75 = 0 \] \[ t^2 - 2t - 15 = 0 \implies (t - 5)(t + 3) = 0 \] Since time cannot be negative, \(t = 5\) s. During these 5 seconds, the balloon continues to move up at 10 m/s: \[ \Delta H_{balloon} = 10 \times 5 = 50 \text{ m} \] Total height of the balloon when the stone hits the ground: \[ H = 75 + 50 = 125 \text{ m} \]
Step 4: Final Answer:
The height of the balloon is 125 m.
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