A shell of mass 5M, acted upon by no external force and initially at rest, bursts into three fragments of masses M, 2M and 2M respectively. The first two fragments move in opposite directions with velocities of magnitude 2v and v respectively. The third fragment will

Options

A. move with a velocity v in a direction perpendicular to the other two
B. move with a velocity 2v in the direction of velocity of the first fragment
C. be at rest
D. move with a velocity v in the direction of velocity of the second fragment.\

Answer

To solve the problem, we will use the principle of conservation of momentum. Here’s the step-by-step solution:

### Step 1: Understand the Initial Conditions
- The shell has a mass of \( 5M \) and is initially at rest.
- Therefore, the initial momentum \( P_{\text{initial}} = 0 \).

### Step 2: Identify the Masses and Velocities of the Fragments
- After bursting, the shell breaks into three fragments with the following masses:
- Fragment 1: mass \( M \) moving with velocity \( 2v \) (let's assume to the right).
- Fragment 2: mass \( 2M \) moving with velocity \( v \) (to the left).
- Fragment 3: mass \( 2M \) with an unknown velocity \( v' \).

### Step 3: Set Up the Conservation of Momentum Equation
- According to the conservation of momentum:
\[ P_{\text{initial}} = P_{\text{final}} \]
- Since \( P_{\text{initial}} = 0 \), we have:
\[ 0 = P_{\text{final}} \]
- The final momentum can be expressed as:
\[ P_{\text{final}} = (M \cdot 2v) + (2M \cdot (-v)) + (2M \cdot v') \]
Here, we take the right direction as positive and the left direction as negative.

### Step 4: Substitute Values into the Equation
- Substitute the known values into the momentum equation:
\[ 0 = (M \cdot 2v) + (2M \cdot (-v)) + (2M \cdot v') \]
Simplifying this gives:
\[ 0 = 2Mv - 2Mv + 2Mv' \]
\[ 0 = 2Mv' \]

### Step 5: Solve for the Unknown Velocity \( v' \)
- From the equation \( 2Mv' = 0 \), we can conclude:
\[ v' = 0 \]
- This means that the third fragment is at rest.

### Conclusion
The third fragment does not move; its velocity \( v' \) is zero.

### Answer
The third fragment will be at rest.

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