The momentum of a body is p and its kinetic energy is E. Its momentum becomes 2p. Its kinetic energy will be

Options

A. `(E )/(2)`
B. 3E
C. 2E
D. 4E

Explanation

To solve the problem, we need to relate the kinetic energy of a body to its momentum. Let's break down the solution step by step.

### Step 1: Understand the relationship between momentum and kinetic energy
The momentum \( p \) of a body is given by the formula:
\[ p = mv \]
where \( m \) is the mass of the body and \( v \) is its velocity.

The kinetic energy \( E \) of a body is given by the formula:
\[ E = \frac{1}{2} mv^2 \]

### Step 2: Express kinetic energy in terms of momentum
We can express kinetic energy in terms of momentum. From the momentum formula, we can solve for velocity \( v \):
\[ v = \frac{p}{m} \]

Now, substitute this expression for \( v \) into the kinetic energy formula:
\[ E = \frac{1}{2} m \left(\frac{p}{m}\right)^2 \]
\[ E = \frac{1}{2} m \cdot \frac{p^2}{m^2} \]
\[ E = \frac{p^2}{2m} \]

### Step 3: Analyze the change in momentum
Initially, the momentum is \( p \) and the kinetic energy is \( E \). When the momentum becomes \( 2p \), we need to find the new kinetic energy \( E' \).

### Step 4: Calculate the new kinetic energy
Using the same relationship we derived, we can express the new kinetic energy \( E' \) when the momentum is \( 2p \):
\[ E' = \frac{(2p)^2}{2m} \]
\[ E' = \frac{4p^2}{2m} \]
\[ E' = \frac{4}{2} \cdot \frac{p^2}{m} = 2 \cdot \frac{p^2}{m} \]

### Step 5: Relate the new kinetic energy to the initial kinetic energy
From our earlier expression for \( E \):
\[ E = \frac{p^2}{2m} \]

Now, we can express \( E' \) in terms of \( E \):
\[ E' = 4E \]

### Conclusion
Thus, when the momentum becomes \( 2p \), the new kinetic energy \( E' \) will be:
\[ E' = 4E \]

### Final Answer
The kinetic energy will be \( 4E \).

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