Two bodies of masses `m_(1) and m_(2)` have same kinetic energy. The ratio of their momentum is

Options

A. `sqrt((m_(2))/(m_(1)))`
B. `sqrt((m_(1))/(m_(2)))`
C. `(m_(1)^(2))/(m_(2)^(2))`
D. `(m_(2)^(2))/(m_(1)^(2))`

Answer

To find the ratio of the momentum of two bodies with the same kinetic energy, we can follow these steps:

### Step 1: Write the formula for kinetic energy
The kinetic energy (KE) of an object is given by the formula:
\[ KE = \frac{1}{2} m v^2 \]
where \( m \) is the mass and \( v \) is the velocity of the object.

### Step 2: Express velocity in terms of momentum
Momentum \( p \) is defined as:
\[ p = mv \]
From this, we can express velocity \( v \) as:
\[ v = \frac{p}{m} \]

### Step 3: Substitute velocity into the kinetic energy formula
Substituting \( v \) in the kinetic energy formula, we get:
\[ KE = \frac{1}{2} m \left(\frac{p}{m}\right)^2 = \frac{1}{2} m \frac{p^2}{m^2} = \frac{p^2}{2m} \]

### Step 4: Set the kinetic energies equal
Let the kinetic energies of the two bodies be equal:
\[ \frac{p_1^2}{2m_1} = \frac{p_2^2}{2m_2} \]

### Step 5: Simplify the equation
Cancel \( \frac{1}{2} \) from both sides:
\[ \frac{p_1^2}{m_1} = \frac{p_2^2}{m_2} \]

### Step 6: Rearrange to find the ratio of momentum
Rearranging gives:
\[ \frac{p_1^2}{p_2^2} = \frac{m_1}{m_2} \]
Taking the square root of both sides, we find:
\[ \frac{p_1}{p_2} = \sqrt{\frac{m_1}{m_2}} \]

### Conclusion
Thus, the ratio of the momentum of the two bodies is:
\[ \frac{p_1}{p_2} = \sqrt{\frac{m_1}{m_2}} \]