P, Q and R are three balloons ascending with velocities U, 4U and 8U respectively. If stones of the same mass be dropped from each, when they are at the same height, then

Options

A. ` P
B. Q
C. R
D. Three stones take equal time

Explanation

To solve the problem, we need to analyze the situation where three balloons (P, Q, and R) are ascending with different velocities and stones are dropped from each balloon at the same height. We will determine which stone reaches the ground first.

### Step-by-Step Solution:

1. **Identify the velocities of the balloons**:
- Balloon P ascends with velocity \( U \).
- Balloon Q ascends with velocity \( 4U \).
- Balloon R ascends with velocity \( 8U \).

2. **Define the motion of the stones**:
- When a stone is dropped from a balloon, it will have an initial upward velocity equal to the velocity of the balloon at the moment of release.
- Therefore, the initial velocities of the stones when dropped are:
- Stone from P: \( U \)
- Stone from Q: \( 4U \)
- Stone from R: \( 8U \)

3. **Set up the equations of motion**:
- The stones are subject to gravitational acceleration \( g \) acting downwards. We will use the equation of motion:
\[ h = ut + \frac{1}{2}(-g)t^2 \]
- Here, \( h \) is the height from which the stones are dropped (which we will take as negative since it is downward), \( u \) is the initial velocity of the stone, and \( t \) is the time taken to reach the ground.

4. **Write the equations for each stone**:
- For the stone from balloon P:
\[ -h = Ut - \frac{1}{2}gt^2 \]
- For the stone from balloon Q:
\[ -h = 4Ut - \frac{1}{2}gt^2 \]
- For the stone from balloon R:
\[ -h = 8Ut - \frac{1}{2}gt^2 \]

5. **Rearranging the equations**:
- Rearranging gives us:
\[ \frac{1}{2}gt^2 - Ut = h \quad \text{(for P)} \]
\[ \frac{1}{2}gt^2 - 4Ut = h \quad \text{(for Q)} \]
\[ \frac{1}{2}gt^2 - 8Ut = h \quad \text{(for R)} \]

6. **Solve for time \( t \)**:
- For each equation, we can solve for \( t \) using the quadratic formula. The general form is:
\[ at^2 + bt + c = 0 \]
- For the stone from P:
\[ \frac{1}{2}g t^2 - Ut + h = 0 \]
- The roots will give us the time taken for each stone to reach the ground.

7. **Compare the times**:
- The time taken \( t \) is inversely related to the initial velocity of the stone. The higher the initial velocity, the longer it will take to reach the ground.
- Since \( U < 4U < 8U \), we conclude:
- \( t_P < t_Q < t_R \)

8. **Conclusion**:
- The stone dropped from balloon P will reach the ground first, followed by the stone from Q, and finally the stone from R.

### Final Answer:
The stone dropped from balloon P reaches the ground first.