If a ball is thrown upwards from the surface of earth :

Options

A. The earth remains stationary while the ball moves upwards
B. The ball remains stationary while the earth moves downwards
C. The ball and earth both move towards each other
D. The ball and earth both move away from each other

Answer

To solve the problem of a ball thrown upwards from the surface of the Earth, we can analyze the situation using the principles of conservation of momentum. Here’s a step-by-step solution:

### Step 1: Understand the System
When a ball is thrown upwards from the surface of the Earth, we have two objects in the system: the ball and the Earth. The ball has a mass \( m_1 \) and is thrown with an initial velocity \( v_1 \). The mass of the Earth is \( M \).

**Hint:** Identify the two objects involved and their properties (mass and initial velocity).

### Step 2: Apply Conservation of Momentum
Since there are no external forces acting on the system (the ball and the Earth), the total linear momentum of the system before and after the ball is thrown must be conserved.

Before the ball is thrown, the total momentum is:
\[ P_{\text{initial}} = 0 \]
(since both the ball and the Earth are stationary).

After the ball is thrown, the total momentum is:
\[ P_{\text{final}} = m_1 v_1 + M v \]
where \( v \) is the velocity of the Earth after the ball is thrown.

**Hint:** Remember that the total momentum before and after must be equal.

### Step 3: Set Up the Equation
Using the conservation of momentum, we can set up the equation:
\[ 0 = m_1 v_1 + M v \]
Rearranging this gives:
\[ M v = -m_1 v_1 \]

**Hint:** Rearranging the equation helps isolate the velocity of the Earth.

### Step 4: Solve for the Velocity of the Earth
From the equation \( M v = -m_1 v_1 \), we can solve for the velocity of the Earth:
\[ v = -\frac{m_1 v_1}{M} \]

**Hint:** The negative sign indicates that the direction of the Earth's velocity is opposite to that of the ball.

### Step 5: Analyze the Directions
Since the ball is moving upwards (in the positive direction), the negative sign for \( v \) indicates that the Earth is moving downwards (in the negative direction). Thus, the ball and the Earth are moving away from each other.

**Hint:** Consider the direction of motion for both the ball and the Earth to conclude their relative movement.

### Conclusion
The correct interpretation of the scenario is that the ball and the Earth both move away from each other after the ball is thrown upwards.

**Final Answer:** The correct option is that the ball and the Earth both move away from each other.