Two masses of 1 kg and 16 kg are moving with equal kinetic energy. The ratio of magnitude of the linear momentum is:

Options

A. `1 : 2`
B. `1 : 4`
C. `1 : sqrt(2)`
D. `sqrt(2) : 1`

Explanation

To solve the problem, we need to find the ratio of the linear momentum of two masses (1 kg and 16 kg) that are moving with equal kinetic energy. Let's break down the solution step by step.

### Step 1: 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 2: Set up the equations for both masses
Let:
- Mass \(m_1 = 1 \, \text{kg}\) with velocity \(v_1\)
- Mass \(m_2 = 16 \, \text{kg}\) with velocity \(v_2\)

The kinetic energy for both masses can be expressed as:
\[ KE_1 = \frac{1}{2} m_1 v_1^2 = \frac{1}{2} (1) v_1^2 = \frac{1}{2} v_1^2 \]
\[ KE_2 = \frac{1}{2} m_2 v_2^2 = \frac{1}{2} (16) v_2^2 = 8 v_2^2 \]

### Step 3: Set the kinetic energies equal to each other
Since both masses have equal kinetic energy:
\[ KE_1 = KE_2 \]
Thus,
\[ \frac{1}{2} v_1^2 = 8 v_2^2 \]

### Step 4: Solve for the relationship between \(v_1\) and \(v_2\)
Multiply both sides by 2 to eliminate the fraction:
\[ v_1^2 = 16 v_2^2 \]
Taking the square root of both sides gives:
\[ v_1 = 4 v_2 \]

### Step 5: Write the expressions for linear momentum
The linear momentum \(p\) is given by:
\[ p = mv \]
For the two masses, we have:
\[ p_1 = m_1 v_1 = 1 \cdot v_1 = v_1 \]
\[ p_2 = m_2 v_2 = 16 \cdot v_2 \]

### Step 6: Find the ratio of linear momentum
Now, we can find the ratio of their linear momentum:
\[ \frac{p_1}{p_2} = \frac{v_1}{16 v_2} \]
Substituting \(v_1 = 4 v_2\) into the equation gives:
\[ \frac{p_1}{p_2} = \frac{4 v_2}{16 v_2} \]
The \(v_2\) cancels out:
\[ \frac{p_1}{p_2} = \frac{4}{16} = \frac{1}{4} \]

### Conclusion
The ratio of the magnitudes of the linear momentum of the two masses is:
\[ 1:4 \]