A stone is dropped from a balloon going up with a uniform velocity of 5 m/sec. If the balloon was 50 m high, then the stone was dropped, the height of balloon from ground when stone hits the ground will be : (g = 10 m/s$^2$)

Question

Accepted Answer

To determine the height of the balloon from the ground when the stone hits the ground, we need to analyze the motion of both the stone and the balloon.

1. **Initial Conditions:**
   - The balloon is moving upwards with a uniform velocity of 5 m/s.
   - The stone is dropped from a height of 50 m above the ground.
   - Acceleration due to gravity, $ g = 10 \, \text{m/s}^2 $.

2. **Motion of the Stone:**
   - When the stone is dropped, it initially has the same upward velocity as the balloon, which is 5 m/s.
   - The stone will first decelerate due to gravity until its velocity becomes zero, and then it will accelerate downwards.

3. **Time for the Stone to Hit the Ground:**
   - Using the equation of motion: $ s = ut + \frac{1}{2} a t^2 $, where:
     - $ s $ is the displacement (50 m, downward),
     - $ u $ is the initial velocity (5 m/s, upward),
     - $ a $ is the acceleration (-10 m/s$^2$, since gravity acts downward).

- The equation becomes: $ -50 = 5t - \frac{1}{2} \times 10 \times t^2 $.
   - Simplifying: $ -50 = 5t - 5t^2 $.
   - Rearranging: $ 5t^2 - 5t - 50 = 0 $.
   - Dividing by 5: $ t^2 - t - 10 = 0 $.

- Solving this quadratic equation using the quadratic formula $ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $, where $ a = 1, b = -1, c = -10 $:
     - $ t = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \times 1 \times (-10)}}{2 \times 1} $.
     - $ t = \frac{1 \pm \sqrt{1 + 40}}{2} $.
     - $ t = \frac{1 \pm \sqrt{41}}{2} $.

- Calculating the positive root (since time cannot be negative):
     - $ t \approx \frac{1 + 6.4}{2} \approx 3.7 \, \text{seconds} $.

4. **Height of the Balloon When Stone Hits the Ground:**
   - The balloon continues to move upwards with a constant velocity of 5 m/s.
   - In 3.7 seconds, the balloon ascends an additional height: $ \text{Height} = \text{velocity} \times \text{time} = 5 \times 3.7 = 18.5 \, \text{m} $.

- Therefore, the total height of the balloon from the ground when the stone hits the ground is:
     - Initial height + Additional height = 50 m + 18.5 m = 68.5 m.

Thus, the correct answer is Option 3: 68.5 m.

**Explanation for Incorrect Options:**

- **Option 1 (63.5 m):** This option underestimates the additional height gained by the balloon during the time the stone takes to hit the ground.
- **Option 2 (65.5 m):** This option also underestimates the additional height gained by the balloon.
- **Option 4 (None of these):** This option is incorrect because we have shown through calculations that the correct height is indeed 68.5 m, which matches Option 3.

Therefore, Option 3 is the correct answer, as it accurately reflects the height of the balloon when the stone hits the ground.