To solve the problem, we need to use Newton's second law of motion, which states that the force acting on an object is equal to the mass of that object multiplied by its acceleration. This can be expressed with the formula: \[ F = m \cdot a \] Where: - \( F \) is the force, - \( m \) is the mass, - \( a \) is the acceleration. ### Step-by-Step Explanation 1. **Identify the Masses and Acceleration**: - Let the mass of the first body be \( m_1 = 3x \) (where \( x \) is a common factor). - Let the mass of the second body be \( m_2 = x \). - The acceleration of the first body is \( a_1 = 2y \) (where \( y \) is another common factor). - The acceleration of the second body is \( a_2 = 9y \). 2. **Calculate the Forces**: - For the first body, using Newton's second law: \[ F_1 = m_1 \cdot a_1 = (3x) \cdot (2y) = 6xy \] - For the second body: \[ F_2 = m_2 \cdot a_2 = (x) \cdot (9y) = 9xy \] 3. **Find the Ratio of Forces**: - Now, we need to find the ratio of the forces \( F_1 \) to \( F_2 \): \[ \text{Ratio of forces} = \frac{F_1}{F_2} = \frac{6xy}{9xy} \] - The \( xy \) terms cancel out: \[ \frac{F_1}{F_2} = \frac{6}{9} = \frac{2}{3} \] 4. **Final Ratio**: - Therefore, the ratio of the forces experienced by the two masses is \( 2:3 \). ### Explanation of Other Options - **Option A: 1 : 4**: This option suggests that the force on the first body is much smaller compared to the second body, which contradicts our calculations showing that the first body experiences a force of \( 6xy \) while the second experiences \( 9xy \). - **Option B: 2 : 1**: This option implies that the first body experiences twice the force of the second body, which is incorrect based on our calculations. The first body has a force of \( 6xy \) and the second \( 9xy \). - **Option D: 2 : 5**: This option also does not match our calculated ratio. The forces are not in this proportion, as we found \( 2:3 \). ### Common Pitfalls - **Ignoring Units**: Always ensure that the units of mass and acceleration are consistent. In this case, we used arbitrary factors \( x \) and \( y \) to simplify the calculations. - **Misinterpreting Ratios**: Be careful when interpreting ratios; they must be derived from the correct relationships between mass and acceleration. ### Revision Summary - Use Newton's second law \( F = m \cdot a \) to relate force, mass, and acceleration. - Calculate forces for each body based on their respective masses and accelerations. - Simplify the ratio of forces carefully, ensuring all terms are accounted for. - Check each option against your calculated result to identify the correct answer. The correct answer is **C: 2 : 3**.