Two masses 1g and 9g are moving with equal kinetic energies. The ratio of the magnitudes of their respective linear momenta is

Options

A. `1:9`
B. `9:1`
C. `1:3`
D. `3:1`

Answer

To solve the problem of finding the ratio of the magnitudes of the linear momenta of two masses (1g and 9g) moving with equal kinetic energies, we can follow these steps:

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

The linear momentum (P) of an object is given by:
\[ P = mv \]

### Step 2: Express kinetic energy in terms of momentum
We can express the kinetic energy in terms of momentum. Since \( P = mv \), we can rearrange this to find \( v \):
\[ v = \frac{P}{m} \]

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: Set the kinetic energies equal for both masses
According to the problem, the kinetic energies of both masses are equal:
\[ KE_1 = KE_2 \]

Using the expression derived above, we can write:
\[ \frac{P_1^2}{2m_1} = \frac{P_2^2}{2m_2} \]

### Step 4: Simplify the equation
The factor of \( \frac{1}{2} \) cancels out:
\[ \frac{P_1^2}{m_1} = \frac{P_2^2}{m_2} \]

### Step 5: Rearrange to find the ratio of momenta
Rearranging gives:
\[ \frac{P_1^2}{P_2^2} = \frac{m_1}{m_2} \]

Taking the square root of both sides yields:
\[ \frac{P_1}{P_2} = \sqrt{\frac{m_1}{m_2}} \]

### Step 6: Substitute the given masses
Given that \( m_1 = 1 \text{ g} \) and \( m_2 = 9 \text{ g} \):
\[ \frac{P_1}{P_2} = \sqrt{\frac{1}{9}} = \frac{1}{3} \]

### Conclusion
Thus, the ratio of the magnitudes of their respective linear momenta is:
\[ \frac{P_1}{P_2} = \frac{1}{3} \]