Trees at the bottom of a lake are approximated as a truncated cone with a base diameter of 6 ft, a top diameter of 1 ft, and a height of 75 ft were tied down with a rope with a vertical force of 20000 lbf. Let's now consider the case when the rope is cut. The trees will quickly rise towards the surface, slowed by the drag of water at high velocities, and fly out of the water to some (substantial) height. Let us assume the force of drag is:
Fd =1/2*p_water*v^2*cdA where cd = 0.40, A = (pi)(3 ft)2, and v is the vertical velocity.
(a): What terminal velocity will the tree reach on it's trip to the surface? Assume that the depth of Dam is sufficient for the tree to reach a terminal velocity. (hint: consider the free body diagram after the rope is cut and the tree is moving. What forces must balance?)
(b): What happens to the total energy of the tree as it rises from lower floor of the damn to the water surface, slowed by the drag force? Does it increase or decrease? Where does the energy go?
(c): If we remove both the forces of drag and buoyancy the moment the tip of the tree breaks the surface, how high will the tree fly? What is a better way to treat the buoyancy force near the surface?