Two masses 10 gm and 40 gm are moving with kinetic energies in the ratio 9:25. Theratio of their linear momenta is

Options

A. `5:6`
B. `3:10`
C. `6:5`
D. `10:3`

Answer

To solve the problem of finding the ratio of linear momenta of two masses given their kinetic energy ratio, we can follow these steps:

### Step 1: Understand the given data
We have two masses:
- Mass \( m_1 = 10 \, \text{g} \)
- Mass \( m_2 = 40 \, \text{g} \)

The kinetic energies are in the ratio:
\[ KE_1 : KE_2 = 9 : 25 \]

### Step 2: Write the formula for kinetic energy
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.

### Step 3: Set up the equation based on the kinetic energy ratio
From the ratio of kinetic energies, we can write:
\[ \frac{KE_1}{KE_2} = \frac{\frac{1}{2} m_1 v_1^2}{\frac{1}{2} m_2 v_2^2} = \frac{m_1 v_1^2}{m_2 v_2^2} \]
Given that \( \frac{KE_1}{KE_2} = \frac{9}{25} \), we can substitute this into our equation:
\[ \frac{10 \cdot v_1^2}{40 \cdot v_2^2} = \frac{9}{25} \]

### Step 4: Simplify the equation
We can simplify the left side:
\[ \frac{10}{40} \cdot \frac{v_1^2}{v_2^2} = \frac{1}{4} \cdot \frac{v_1^2}{v_2^2} \]
Thus, our equation becomes:
\[ \frac{1}{4} \cdot \frac{v_1^2}{v_2^2} = \frac{9}{25} \]

### Step 5: Solve for \(\frac{v_1^2}{v_2^2}\)
To isolate \(\frac{v_1^2}{v_2^2}\), we multiply both sides by 4:
\[ \frac{v_1^2}{v_2^2} = \frac{9 \cdot 4}{25} = \frac{36}{25} \]

### Step 6: Take the square root to find the ratio of velocities
Taking the square root of both sides gives us:
\[ \frac{v_1}{v_2} = \frac{6}{5} \]

### Step 7: Write the formula for linear momentum
The linear momentum \( p \) is given by:
\[ p = mv \]
Thus, the momenta of the two masses are:
\[ p_1 = m_1 v_1 \quad \text{and} \quad p_2 = m_2 v_2 \]

### Step 8: Find the ratio of linear momenta
The ratio of their momenta is:
\[ \frac{p_1}{p_2} = \frac{m_1 v_1}{m_2 v_2} = \frac{10 \cdot v_1}{40 \cdot v_2} \]
Substituting \(\frac{v_1}{v_2} = \frac{6}{5}\):
\[ \frac{p_1}{p_2} = \frac{10 \cdot \frac{6}{5}}{40} = \frac{60}{200} = \frac{3}{10} \]

### Final Answer
The ratio of their linear momenta is:
\[ \frac{p_1}{p_2} = 3 : 10 \]