Two masses of 1 gm and 4 gm are moving with equal kinetic energies. The ratio of the magnitudes of their linear momenta is

Options

A. `4:1`
B. `sqrt2:1`
C. `1:2`
D. `1:16`

Explanation

To find the ratio of the magnitudes of the linear momenta of two masses (1 gm and 4 gm) 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:
\[ KE = \frac{p^2}{2m} \]
where \( p \) is the momentum and \( m \) is the mass of the object.

### Step 2: Set up the equations for both masses
Let the first mass \( m_1 = 1 \, \text{g} \) and the second mass \( m_2 = 4 \, \text{g} \). Since both masses have equal kinetic energies, we can write:
\[ KE_1 = KE_2 \]
This gives us:
\[ \frac{p_1^2}{2m_1} = \frac{p_2^2}{2m_2} \]

### Step 3: Simplify the equation
We can cancel the factor of \( \frac{1}{2} \) from both sides:
\[ \frac{p_1^2}{m_1} = \frac{p_2^2}{m_2} \]

### Step 4: Rearrange to find the ratio of momenta
Cross-multiplying gives us:
\[ p_1^2 \cdot m_2 = p_2^2 \cdot m_1 \]
Now, we can express the ratio of the squares of the momenta:
\[ \frac{p_1^2}{p_2^2} = \frac{m_1}{m_2} \]

### Step 5: Substitute the values of the masses
Substituting \( m_1 = 1 \, \text{g} \) and \( m_2 = 4 \, \text{g} \):
\[ \frac{p_1^2}{p_2^2} = \frac{1}{4} \]

### Step 6: Take the square root to find the ratio of momenta
Taking the square root of both sides gives us:
\[ \frac{p_1}{p_2} = \sqrt{\frac{1}{4}} = \frac{1}{2} \]

### Step 7: Write the final ratio
Thus, the ratio of the magnitudes of their linear momenta is:
\[ p_1 : p_2 = 1 : 2 \]

### Final Answer
The ratio of the magnitudes of their linear momenta is \( 1 : 2 \).

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