Two stones are thrown vertically upwards simultaneously from the same point on the ground with initial speed `u_(1) = 30 m//sec` and `u_(2) = 50 m//sec`. Which of the curve represent correct variation (for the time interval in which both reach the ground) of
`(x_(2) - x_(1))` = the relative position of second stone with respect to first with time (t)
`(v_(2) - v_(1))` = the relative velocity of second stone with respect to first with time (t).
Assuming that stones do not rebound after hitting.

Options

A.
B.
C.
D.

Explanation

To solve the problem, we need to analyze the motion of the two stones thrown vertically upwards with different initial velocities. We will derive the expressions for the relative position and relative velocity of the two stones over time.

### Step 1: Determine the equations of motion for both stones.

The equations of motion for an object thrown vertically upwards are given by:
\[ x(t) = ut + \frac{1}{2} a t^2 \]
where \( u \) is the initial velocity, \( a \) is the acceleration (which is \(-g\) for upward motion, where \( g \approx 9.81 \, \text{m/s}^2 \)), and \( t \) is the time.

For the first stone (with initial speed \( u_1 = 30 \, \text{m/s} \)):
\[ x_1(t) = u_1 t - \frac{1}{2} g t^2 = 30t - \frac{1}{2} (9.81) t^2 \]

For the second stone (with initial speed \( u_2 = 50 \, \text{m/s} \)):
\[ x_2(t) = u_2 t - \frac{1}{2} g t^2 = 50t - \frac{1}{2} (9.81) t^2 \]

### Step 2: Calculate the relative position \( x_2 - x_1 \).

Now, we find the relative position of the second stone with respect to the first:
\[ x_2 - x_1 = (50t - \frac{1}{2} (9.81) t^2) - (30t - \frac{1}{2} (9.81) t^2) \]
\[ x_2 - x_1 = (50t - 30t) = 20t \]

### Step 3: Analyze the relative position over time.

The relative position \( x_2 - x_1 \) is directly proportional to time \( t \). This means that as time increases, the difference in position increases linearly.

### Step 4: Determine the equations for the velocities.

The velocities of the stones at any time \( t \) can be calculated using:
\[ v(t) = u + at \]

For the first stone:
\[ v_1(t) = u_1 - gt = 30 - 9.81t \]

For the second stone:
\[ v_2(t) = u_2 - gt = 50 - 9.81t \]

### Step 5: Calculate the relative velocity \( v_2 - v_1 \).

Now, we find the relative velocity of the second stone with respect to the first:
\[ v_2 - v_1 = (50 - 9.81t) - (30 - 9.81t) \]
\[ v_2 - v_1 = 50 - 30 = 20 \]

### Step 6: Analyze the relative velocity over time.

The relative velocity \( v_2 - v_1 \) is constant at \( 20 \, \text{m/s} \). This means that the difference in velocities does not change over time.

### Conclusion:

1. The relative position \( x_2 - x_1 \) varies linearly with time \( t \) and can be represented by a straight line graph.
2. The relative velocity \( v_2 - v_1 \) is constant over time and can be represented by a horizontal line graph.