How does a sound wave transfer energy to your ears?
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Jane's mechanical energy at any time is
where
is the potential energy, while 
is the kinetic energy.
Initially, Jane is on the ground, so the altitude is h=0 and the potential energy is zero: U=0. She's running with speed v, so she has kinetic energy only:
Then she grabs the vine, and when she reaches the maximum height h, her speed is zero: v=0, and so the kinetic energy becomes zero: K=0. So now her mechanical energy is just potential energy:
But E must be conserved, so the initial kinetic energy must be equal to the final potential energy:
from which we can find h, the maximum height Jane can reach:
where
Initially, Jane is on the ground, so the altitude is h=0 and the potential energy is zero: U=0. She's running with speed v, so she has kinetic energy only:
Then she grabs the vine, and when she reaches the maximum height h, her speed is zero: v=0, and so the kinetic energy becomes zero: K=0. So now her mechanical energy is just potential energy:
But E must be conserved, so the initial kinetic energy must be equal to the final potential energy:
from which we can find h, the maximum height Jane can reach:
Refer to the diagram shown below.
h = original height of the pelican when the fish is dropped (not relevant).
S = distance traveled by the fish as a function of time, measured upward.
u = 0.5 m/s, the upward velocity with which the fish is dropped.
g = 9.8 m/s², the acceleration due to gravity.
Use the following equation:
S = ut + (1/2)gt²
S = (0.5 m/s)*(2.5 s) + 0.5*(-9.8 m/s²)*(2.5 s)²
= -29.375 m
The negative sign means that the fish drops by 29.375 m from the original height of h.
Answer: The fish is 29.375 m below where the pelican dropped it after 2.5 s.
h = original height of the pelican when the fish is dropped (not relevant).
S = distance traveled by the fish as a function of time, measured upward.
u = 0.5 m/s, the upward velocity with which the fish is dropped.
g = 9.8 m/s², the acceleration due to gravity.
Use the following equation:
S = ut + (1/2)gt²
S = (0.5 m/s)*(2.5 s) + 0.5*(-9.8 m/s²)*(2.5 s)²
= -29.375 m
The negative sign means that the fish drops by 29.375 m from the original height of h.
Answer: The fish is 29.375 m below where the pelican dropped it after 2.5 s.