Final answer:
The terminal velocity of a bowling ball dropped in water is calculated by setting the sum of buoyant and drag forces equal to the gravitational force. Solving the quadratic equation for velocity provides the terminal speed. Calculating the time to hit the bottom usually requires numerical methods due to the changing velocity influenced by drag and buoyancy.
Step-by-step explanation:
The question involves calculating the terminal velocity of a bowling ball dropped in water, considering drag and buoyancy forces, and determining the time it takes to reach the bottom of a body of water.
To find terminal velocity, we use the equilibrium condition where the sum of all forces equals zero (F_net = 0), because at terminal velocity, acceleration is zero. The forces acting on the ball are gravity (downward), buoyancy (upward), and drag (upward).
The gravitational force (weight) is given by Fg = mg, where m is the mass of the ball, and g is the acceleration due to gravity. The buoyant force is calculated using Archimedes' principle: Fb = ρ_water * V_ball * g, where ρ_water is the density of water, V_ball is the volume of the ball, and g is the gravitational acceleration. The drag force, according to the drag equation, is Fd = 0.5 * C_d * ρ_water * A * v^2, where C_d is the drag coefficient, A is the cross-sectional area, and v is the velocity.
Setting the sum of the buoyant and drag forces equal to the gravitational force, and solving for terminal velocity (v_terminal), we get the quadratic equation to solve for v. Finally, to find the time to reach the bottom, we would integrate the motion equation considering the changing velocity due to drag and buoyancy, which is a more complex calculation often requiring numerical methods or software.