Chapter 3.4, Problem 10E

Unless otherwise stated, in the following problems we assume that the gravitational force is constant with $g=9.81{\text{m/sec}}^{\text{2}}$ in the MKS system and $g=32{\text{ft/sec}}^{\text{2}}$ in the U.S. Customary system.

An object of mass $2\text{kg}$ is released from rest from a platform $30\text{m}$ above the water and allowed to fall under the influence of gravity. After the object strikes the water, it begins to sink with gravity pulling down and buoyancy force pushing up. Assume that the force of gravity is constant, no change in momentum occurs on impact with the water, the buoyancy force is $1/2$ the weight ( $\text{weight}=mg$), and the force due to air resistance or water resistance is proportional to the velocity, with the proportionality constant $b_1=10\text{N-m/sec}$ in the air and $b_2=100\text{N-m/sec}$ in the water. Find the equation of motion of the object. What is the velocity of the object $1\min$ after it is released?