Gayle runs at a speed of 9.00 m/s and dives on a sled, initially at rest on the top of a frictionless, snow-covered hill, that has a vertical drop of 20.0 m. After she has descended a vertical distance of 4.00 m, her brother, who is initially at rest, hops on her back, and they continue down the hill together. (d) What is their speed at the bottom of the hill?

Gayle’s mass is 70.0 kg, the sled has a mass of 2.00 kg, and her brother has a mass of 50.0 kg.

What value are you getting? I keep getting the wrong answer.

We are not going to feed answer grazers. Show your work, and we can point out errors.

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

Step 1: Calculate Gayle's initial gravitational potential energy at the top of the hill.
The gravitational potential energy is given by the equation: PE = m * g * h
where m is the mass, g is the acceleration due to gravity (9.8 m/s²), and h is the vertical height.

For Gayle: PE_gayle = m_gayle * g * h
= 70.0 kg * 9.8 m/s² * 20.0 m
= 13720 J

Step 2: Calculate the sled's initial gravitational potential energy at the top of the hill.
Note that the sled starts at rest, so it doesn't have any kinetic energy. Its total initial mechanical energy is equal to its potential energy.

For the sled: PE_sled = m_sled * g * h
= 2.00 kg * 9.8 m/s² * 20.0 m
= 392 J

Step 3: Calculate Gayle and her brother's combined initial gravitational potential energy at the top of the hill.
Since her brother is initially at rest, he doesn't contribute any kinetic energy initially. Therefore, his mass is only considered for gravitational potential energy.

Total PE_initial = PE_gayle + PE_sled
= 13720 J + 392 J
= 14112 J

Step 4: Calculate the final kinetic energy at the bottom of the hill.
Since the hill is frictionless, there is no energy loss due to friction. Therefore, all the initial potential energy at the top of the hill is converted into kinetic energy at the bottom.

Total KE_final = Total PE_initial
= 14112 J

Step 5: Calculate the total mass of Gayle and her brother.
Total mass = m_gayle + m_sled + m_brother
= 70.0 kg + 2.00 kg + 50.0 kg
= 122.0 kg

Step 6: Calculate the final speed using the equation for kinetic energy:
KE = (1/2) * m * v², where v is the final speed.

Total KE_final = (1/2) * m * v²
v² = (2 * Total KE_final) / m
v = √((2 * Total KE_final) / m)

Plugging in the values, we get:
v = √((2 * 14112 J) / 122.0 kg)
= √(28224 J / 122.0 kg)
= √231.541 m²/s²
= 15.21 m/s

Therefore, the speed of Gayle and her brother at the bottom of the hill is approximately 15.21 m/s.