Two stones are thrown up simultaneously from the edge of a cliff with initial speed `v and 2 v`. The relative position of the second stone with respect to first varies with time till both the stones strike the ground as.

Options

A. linearly
B. first linearly then parabolically
C. parabolically
D. first parabolically then linearly

Answer

To solve the problem, we need to analyze the motion of two stones thrown upward from the edge of a cliff with different initial velocities. Let's denote the initial velocity of the first stone as \( v \) and that of the second stone as \( 2v \). We will find the relative position of the second stone with respect to the first stone over time until both stones hit the ground.

### Step 1: Define the Initial Conditions
- Let the first stone be thrown with an initial velocity \( v \).
- Let the second stone be thrown with an initial velocity \( 2v \).
- Both stones are thrown simultaneously from the same height \( h \).

### Step 2: Determine the Relative Position
The relative position of the second stone with respect to the first stone can be expressed as:
\[ s_{21} = s_2 - s_1 \]
where \( s_2 \) is the position of the second stone and \( s_1 \) is the position of the first stone.

### Step 3: Write the Equations of Motion
For the first stone (with initial velocity \( v \)):
\[ s_1 = h + vt - \frac{1}{2}gt^2 \]
For the second stone (with initial velocity \( 2v \)):
\[ s_2 = h + 2vt - \frac{1}{2}gt^2 \]

### Step 4: Calculate the Relative Position
Substituting the equations of motion into the relative position equation:
\[ s_{21} = (h + 2vt - \frac{1}{2}gt^2) - (h + vt - \frac{1}{2}gt^2) \]
This simplifies to:
\[ s_{21} = (2vt - vt) = vt \]

### Step 5: Analyze the Motion
- For the upward motion, the relative position \( s_{21} = vt \) is linear with respect to time \( t \).
- When both stones reach their maximum height and start falling down, the relative acceleration between the two stones becomes significant.

### Step 6: Determine the Downward Motion
During the downward motion, the first stone will have an acceleration of \( g \) (as it falls freely after reaching the maximum height), while the second stone will also have an acceleration of \( g \). Thus, the relative position will change as follows:
\[ s_{21} = vt + \frac{1}{2}gt^2 \]
This indicates that the relative position becomes parabolic as both stones fall under gravity.

### Conclusion
The relative position of the second stone with respect to the first stone varies first linearly (during the upward motion) and then parabolically (during the downward motion) until both stones strike the ground.

### Final Answer
The correct option is: **first linearly then parabolically**.