A spring is compressed between two toy cars. When the cars are released, they move apart. If `x_1 and x_2` are displacements of the cars when in contact with the spring, then (mass ae `m_1 and m_2`)

Options

A. `m_1x_1 = m_2x_2`
B. `m_1x_2 = m_2x_1`
C. `m_1x_(1)^(2) = m_(2)x_(2)^(2)`
D. `m_1x_(2)^(2) = m_(2)x_(1)^(2)`

Answer

To solve the problem of the two toy cars and the spring, we will derive the relationship between the masses of the cars and their respective displacements when the spring is released.

### Step-by-Step Solution:

1. **Understanding the Forces**:
When the spring is released, it exerts equal and opposite forces on both toy cars. Let’s denote the force exerted on car 1 (mass \( m_1 \)) as \( F_1 \) and the force on car 2 (mass \( m_2 \)) as \( F_2 \). According to Newton's third law, \( F_1 = F_2 \).

2. **Applying Newton's Second Law**:
According to Newton's second law, the force acting on an object is equal to the mass of the object multiplied by its acceleration. Therefore, we can write:
\[ F_1 = m_1 \cdot a_1 \quad \text{and} \quad F_2 = m_2 \cdot a_2 \]
Since \( F_1 = F_2 \), we have:
\[ m_1 \cdot a_1 = m_2 \cdot a_2 \quad \text{(Equation 1)} \]

3. **Relating Displacement and Acceleration**:
The displacement \( x \) of each car can be related to its acceleration and time using the equation of motion. Since both cars start from rest, we can express their displacements as:
\[ x_1 = \frac{1}{2} a_1 t^2 \quad \text{and} \quad x_2 = \frac{1}{2} a_2 t^2 \]
From these equations, we can solve for the accelerations:
\[ a_1 = \frac{2x_1}{t^2} \quad \text{and} \quad a_2 = \frac{2x_2}{t^2} \]

4. **Substituting Accelerations into Equation 1**:
Now we substitute the expressions for \( a_1 \) and \( a_2 \) back into Equation 1:
\[ m_1 \cdot \frac{2x_1}{t^2} = m_2 \cdot \frac{2x_2}{t^2} \]
The \( t^2 \) terms cancel out:
\[ m_1 \cdot 2x_1 = m_2 \cdot 2x_2 \]

5. **Simplifying the Equation**:
Dividing both sides by 2 gives us:
\[ m_1 \cdot x_1 = m_2 \cdot x_2 \]

6. **Final Relationship**:
Thus, we have established the relationship between the masses and the displacements:
\[ m_1 x_1 = m_2 x_2 \]

### Conclusion:
The relationship between the masses of the toy cars and their respective displacements when the spring is released is given by:
\[ m_1 x_1 = m_2 x_2 \]