The correct option is **B. weight is equal to the retarding force**. ### Detailed Explanation: When a particle falls through a fluid (like water or air), it experiences several forces acting on it: 1. **Weight (W)**: This is the force due to gravity acting downwards on the particle. It can be calculated using the formula: \[ W = mg \] where \( m \) is the mass of the particle and \( g \) is the acceleration due to gravity. 2. **Buoyant Force (B)**: This is the upward force exerted by the fluid on the particle, which is equal to the weight of the fluid displaced by the particle. According to Archimedes' principle, the buoyant force can be calculated as: \[ B = \rho_f V g \] where \( \rho_f \) is the density of the fluid, \( V \) is the volume of the particle, and \( g \) is the acceleration due to gravity. 3. **Viscous Retarding Force (F_v)**: This is the force that opposes the motion of the particle through the fluid. It is often proportional to the velocity of the particle and can be described by Stokes' law for small velocities: \[ F_v = 6 \pi \eta r v \] where \( \eta \) is the dynamic viscosity of the fluid, \( r \) is the radius of the particle, and \( v \) is the velocity of the particle. ### Terminal Velocity: As the particle falls, it accelerates due to its weight until it reaches a point where the forces acting on it balance out. This point is known as **terminal velocity**. At terminal velocity, the following condition holds true: \[ W = B + F_v \] When the particle reaches terminal velocity, the net force acting on it is zero, meaning that the downward force (weight) is equal to the sum of the upward forces (buoyant force and viscous retarding force). However, in the context of the options provided, the most relevant statement is that the weight of the particle is equal to the retarding force when considering only the viscous force (assuming buoyancy is negligible or the particle is denser than the fluid). Thus, at terminal velocity, we can simplify this to: \[ W = F_v \] This is why option B is correct: **the weight of the particle is equal to the retarding force** (which is primarily the viscous force at terminal velocity). ### Why Other Options Are Incorrect: - **Option A: Acceleration is maximum** This option is incorrect because, at terminal velocity, the particle no longer accelerates. The acceleration becomes zero as the forces balance out. Therefore, the acceleration is not maximum; it is zero. - **Option C: Buoyancy force is equal to the viscous retarding force** This option is misleading. While the buoyant force does act upwards, at terminal velocity, the weight of the particle is balanced by the sum of the buoyant force and the viscous force. The buoyant force does not equal the viscous force; rather, they work together to balance the weight. - **Option D: Buoyancy force is more than the weight of the fluid displaced** This option is incorrect because, according to Archimedes' principle, the buoyant force is equal to the weight of the fluid displaced. If the buoyant force were greater than the weight of the fluid displaced, the particle would rise instead of falling. ### Summary for Revision: - At terminal velocity, the forces acting on a falling particle in a fluid are balanced. - The weight of the particle equals the retarding force (primarily viscous force) at terminal velocity. - Acceleration is zero at terminal velocity, not maximum. - Buoyant force is equal to the weight of the fluid displaced, not greater or equal to the viscous force. This understanding of forces and terminal velocity is crucial for solving problems related to motion in fluids.