Find the ratio of the linear momenta of two particles of masses 1.0 kg and 4.0 kg if their kinetic energies are equal.

Explanation

To find the ratio of the linear momenta of two particles with masses 1.0 kg and 4.0 kg, given that their kinetic energies are equal, we can follow these steps:

### Step 1: Write down the formula for kinetic energy.
The kinetic energy (KE) of a particle 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 the kinetic energies of both particles.
Let the masses of the two particles be:
- \( m_1 = 1.0 \, \text{kg} \)
- \( m_2 = 4.0 \, \text{kg} \)

Since their kinetic energies are equal, we can write:
\[ KE_1 = KE_2 \]
Substituting the kinetic energy formula, we have:
\[ \frac{1}{2} m_1 v_1^2 = \frac{1}{2} m_2 v_2^2 \]

### Step 3: Simplify the equation.
We can cancel the \(\frac{1}{2}\) from both sides:
\[ m_1 v_1^2 = m_2 v_2^2 \]

### Step 4: Substitute the values of the masses.
Substituting \( m_1 = 1.0 \, \text{kg} \) and \( m_2 = 4.0 \, \text{kg} \):
\[ 1.0 \cdot v_1^2 = 4.0 \cdot v_2^2 \]

### Step 5: Rearrange to find the ratio of velocities.
Rearranging gives:
\[ v_1^2 = 4.0 \cdot v_2^2 \]
Taking the square root of both sides:
\[ v_1 = 2 \cdot v_2 \]

### Step 6: Write the formula for linear momentum.
The linear momentum \( p \) of a particle is given by:
\[ p = mv \]
Thus, for both particles, we have:
- \( p_1 = m_1 v_1 \)
- \( p_2 = m_2 v_2 \)

### Step 7: Substitute the values into the momentum equations.
Substituting the values:
\[ p_1 = 1.0 \cdot v_1 = 1.0 \cdot (2 \cdot v_2) = 2.0 \cdot v_2 \]
\[ p_2 = 4.0 \cdot v_2 \]

### Step 8: Find the ratio of the momenta.
Now, we can find the ratio of the momenta:
\[ \frac{p_1}{p_2} = \frac{2.0 \cdot v_2}{4.0 \cdot v_2} = \frac{2.0}{4.0} = \frac{1}{2} \]

### Final Answer:
Thus, the ratio of the linear momenta of the two particles is:
\[ \text{Ratio} = 1 : 2 \]

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