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. `sqrt(2):1`
C. `1:2`
D. `1:16`

Answer

To solve the problem, we need to find the ratio of the magnitudes of the linear momenta of two masses moving with equal kinetic energies. Let's break down the solution step by step.

### Step 1: Understand the Given Information
We have two masses:
- Mass \( m_1 = 1 \, \text{gm} \) (which is \( 0.001 \, \text{kg} \))
- Mass \( m_2 = 4 \, \text{gm} \) (which is \( 0.004 \, \text{kg} \))

Both masses are moving with equal kinetic energies.

### Step 2: Write the Kinetic Energy Equation
The kinetic energy (KE) of an object is given by the formula:
\[ KE = \frac{1}{2} mv^2 \]
Since both masses have equal kinetic energies, we can write:
\[ \frac{1}{2} m_1 v_1^2 = \frac{1}{2} m_2 v_2^2 \]
We can simplify this equation by canceling \( \frac{1}{2} \) from both sides:
\[ m_1 v_1^2 = m_2 v_2^2 \]

### Step 3: Substitute the Masses
Now, substituting the values of \( m_1 \) and \( m_2 \):
\[ 1 \cdot v_1^2 = 4 \cdot v_2^2 \]
This simplifies to:
\[ v_1^2 = 4 v_2^2 \]

### Step 4: Find the Ratio of Velocities
Taking the square root of both sides gives us:
\[ v_1 = 2 v_2 \]
Thus, the ratio of the velocities is:
\[ \frac{v_1}{v_2} = 2 \]

### Step 5: Write the Formula for Linear Momentum
The linear momentum \( p \) of an object is given by:
\[ p = mv \]
So, for each mass, we have:
- \( p_1 = m_1 v_1 \)
- \( p_2 = m_2 v_2 \)

### Step 6: Find the Ratio of Linear Momenta
Now, we can find the ratio of the magnitudes of their linear momenta:
\[ \frac{p_1}{p_2} = \frac{m_1 v_1}{m_2 v_2} \]
Substituting the values:
\[ \frac{p_1}{p_2} = \frac{1 \cdot v_1}{4 \cdot v_2} \]
Using the ratio of velocities we found earlier (\( v_1 = 2 v_2 \)):
\[ \frac{p_1}{p_2} = \frac{1 \cdot (2 v_2)}{4 \cdot v_2} \]
The \( v_2 \) cancels out:
\[ \frac{p_1}{p_2} = \frac{2}{4} = \frac{1}{2} \]

### Step 7: Conclusion
Thus, the ratio of the magnitudes of their linear momenta is:
\[ \boxed{1:2} \]