A 4 kg mass and a 1 kg mass are moving with equal kinetic energies. The ratio of the magnitudes of their linear momenta is

Options

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

Answer

To solve the problem, we need to find the ratio of the magnitudes of the linear momenta of two masses (4 kg and 1 kg) that are moving with equal kinetic energies.

### Step-by-Step Solution:

1. **Understanding 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.

2. **Setting Up the Equation**:
Let the kinetic energies of the 4 kg mass and the 1 kg mass be equal. Therefore, we can write:
\[ KE_1 = KE_2 \]
This translates to:
\[ \frac{1}{2} m_1 v_1^2 = \frac{1}{2} m_2 v_2^2 \]
where \( m_1 = 4 \, \text{kg} \) and \( m_2 = 1 \, \text{kg} \).

3. **Canceling Out Common Terms**:
The \(\frac{1}{2}\) cancels out from both sides:
\[ m_1 v_1^2 = m_2 v_2^2 \]
Substituting the masses:
\[ 4 v_1^2 = 1 v_2^2 \]

4. **Expressing Velocities**:
Rearranging gives:
\[ v_2^2 = 4 v_1^2 \]
Taking the square root of both sides:
\[ v_2 = 2 v_1 \]

5. **Calculating Linear Momentum**:
The linear momentum \( p \) is given by:
\[ p = mv \]
For the 4 kg mass:
\[ p_1 = m_1 v_1 = 4 v_1 \]
For the 1 kg mass:
\[ p_2 = m_2 v_2 = 1 \cdot (2 v_1) = 2 v_1 \]

6. **Finding the Ratio of Linear Momenta**:
Now, we can find the ratio of the magnitudes of their linear momenta:
\[ \frac{p_1}{p_2} = \frac{4 v_1}{2 v_1} = \frac{4}{2} = 2 \]
Thus, the ratio of the magnitudes of their linear momenta is:
\[ p_1 : p_2 = 2 : 1 \]

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