To determine how the power of a system changes when both the force and the velocity are reduced by half, we need to understand the relationship between power, force, and velocity. ### Step-by-Step Explanation 1. **Understanding Power**: Power (P) is defined as the rate at which work is done or energy is transferred. The formula for power in terms of force (F) and velocity (v) is given by: \[ P = F \cdot v \] This means that power is the product of the force applied to an object and the velocity at which the object moves. 2. **Initial Conditions**: Let's denote the initial force as \( F \) and the initial velocity as \( v \). Therefore, the initial power \( P_{\text{initial}} \) can be expressed as: \[ P_{\text{initial}} = F \cdot v \] 3. **Reducing Force and Velocity**: According to the problem, both the force and the velocity are reduced by half. Thus, the new force \( F' \) and the new velocity \( v' \) can be expressed as: \[ F' = \frac{F}{2} \] \[ v' = \frac{v}{2} \] 4. **Calculating New Power**: Now, we can calculate the new power \( P_{\text{new}} \) using the reduced values of force and velocity: \[ P_{\text{new}} = F' \cdot v' = \left(\frac{F}{2}\right) \cdot \left(\frac{v}{2}\right) \] Simplifying this gives: \[ P_{\text{new}} = \frac{F \cdot v}{4} \] 5. **Comparing Initial and New Power**: Now we can compare the new power to the initial power: \[ P_{\text{new}} = \frac{P_{\text{initial}}}{4} \] This shows that the new power is one-fourth of the initial power. ### Conclusion Thus, when both the force and the velocity are reduced by half, the power of the system is reduced to a quarter of its original value. Therefore, the correct answer is **B. reduced to a quarter**. ### Explanation of Other Options - **A. doubled**: This option is incorrect because reducing both force and velocity leads to a decrease in power, not an increase. - **C. reduced by half**: This option is also incorrect. While reducing either force or velocity by half would reduce power, reducing both results in a more significant reduction (to a quarter). - **D. constant**: This option is incorrect because the power changes when both force and velocity are altered. ### Summary - Power is calculated as the product of force and velocity: \( P = F \cdot v \). - Reducing both force and velocity by half results in power being reduced to a quarter of its original value. - The new power can be calculated as \( P_{\text{new}} = \frac{P_{\text{initial}}}{4} \). - The correct answer is **B. reduced to a quarter**.