Gayle runs at a speed of 3.80 m/s and dives on a sled, initially at rest on the top of a frictionless snow covered hill. After she has descended a vertical distance of 5.00 m, her brother, who is initially at rest, hops on her back and together they continue down the hill. What is their speed at the bottom of the hill if the total vertical drop is 15.0 m? Gayle's mass is 50.0 kg, the sled has a mass of 5.00 kg and her brother has a mass of 30.0 kg.

To find the final speed of Gayle, the sled, and her brother at the bottom of the hill, we can apply the principle of conservation of energy.

1. Calculate the potential energy at the top and bottom of the hill:
- The potential energy at the top (P1) is equal to the initial potential energy of Gayle alone.
P1 = m1 * g * h1
where m1 is Gayle's mass, g is the acceleration due to gravity (approximately 9.8 m/s²), and h1 is the initial vertical distance.
- The potential energy at the bottom (P2) is equal to the sum of the potential energy of Gayle, the sled, and her brother.
P2 = (m1 + m_sled + m_brother) * g * h2
where m_sled is the sled's mass and m_brother is the brother's mass, and h2 is the total vertical drop.

2. Calculate the kinetic energy at the bottom:
- The kinetic energy at the bottom (K2) is equal to the sum of the kinetic energy of Gayle, the sled, and her brother.
K2 = (1/2) * (m1 + m_sled + m_brother) * v^2
where v is the final speed we're trying to find.

3. Apply the conservation of energy principle:
Since energy is conserved, the total energy at the top is equal to the total energy at the bottom (ignoring any energy losses due to friction, air resistance, etc.).
Therefore, P1 = K2.

Substituting the equations and solving for v:
m1 * g * h1 = (1/2) * (m1 + m_sled + m_brother) * v^2

Plugging in the known values:
(50.0 kg) * (9.8 m/s²) * (5.00 m) = (1/2) * (50.0 kg + 5.00 kg + 30.0 kg) * v^2

Now, we can solve for v by isolating it in the equation and taking the square root of both sides:

v^2 = ((50.0 kg) * (9.8 m/s²) * (5.00 m)) / ((1/2) * (50.0 kg + 5.00 kg + 30.0 kg))
v = sqrt(((50.0 kg) * (9.8 m/s²) * (5.00 m)) / ((1/2) * (50.0 kg + 5.00 kg + 30.0 kg)))

Calculating the above expression will give the final speed (v) at the bottom of the hill.