An air bubble is released from rest at the bottom of a tall glass column filled with a highly viscous liquid. As the bubble rises, it experiences a constant buoyant force, its own weight, and a viscous drag force that obeys Stokes' law. Which of the graphs shown below best represents the variation of the bubble's upward acceleration ( a ) as a function of its upward velocity ( v ) until it reaches terminal velocity?

Step-by-step solution

Let the mass of the air bubble be m and its radius be r . The forces acting on the bubble as it rises are: 1. Upward buoyant force, F_B (given as constant) 2. Downward weight, W = mg 3. Downward viscous drag force, F_v According to Stokes' law, the viscous drag force is proportional to the velocity v of the bubble: F_v = 6 r v = kv where k = 6 r is a positive constant. The net upward force F_ net on the bubble is: F_ net = F_B - W - F_v ma = F_B - mg - kv Dividing by m , we get the upward acceleration a as a functi