Kinetic energy of a particle is increases by 4 times What will be the relation between intital and final momentum?

Options

A. `p_(2)=2P_(1)`
B. `P_(2)=p_(1)/2`
C. `p_(2)=P_(1)`
D. `p_(2)=4p_(1)`

Answer

To solve the problem, we need to establish the relationship between kinetic energy and momentum. Let's go through the steps systematically.

### Step 1: Understand the relationship between kinetic energy and momentum
The kinetic energy (KE) of a particle is given by the formula:
\[ KE = \frac{1}{2} mv^2 \]
where \(m\) is the mass of the particle and \(v\) is its velocity.

The momentum (P) of a particle is defined as:
\[ P = mv \]

### Step 2: Express kinetic energy in terms of momentum
From the momentum equation, we can express velocity as:
\[ v = \frac{P}{m} \]
Substituting this into the kinetic energy formula gives:
\[ KE = \frac{1}{2} m \left(\frac{P}{m}\right)^2 = \frac{P^2}{2m} \]

### Step 3: Relate initial and final kinetic energies
Let the initial kinetic energy be \(KE_1\) and the final kinetic energy be \(KE_2\). According to the problem, the final kinetic energy is 4 times the initial kinetic energy:
\[ KE_2 = 4 KE_1 \]

### Step 4: Substitute the kinetic energy expressions
Using the relationship we derived:
\[ KE_1 = \frac{P_1^2}{2m} \quad \text{and} \quad KE_2 = \frac{P_2^2}{2m} \]
Substituting these into the equation \(KE_2 = 4 KE_1\) gives:
\[ \frac{P_2^2}{2m} = 4 \left(\frac{P_1^2}{2m}\right) \]

### Step 5: Simplify the equation
We can cancel \( \frac{1}{2m} \) from both sides:
\[ P_2^2 = 4 P_1^2 \]

### Step 6: Take the square root
Taking the square root of both sides results in:
\[ P_2 = 2 P_1 \]

### Conclusion
Thus, the final momentum \(P_2\) is twice the initial momentum \(P_1\):
\[ P_2 = 2 P_1 \]