A skier leaves the starting gate at the top of a ski ramp with an initial speed of 4.00 m/s (Figure 6-41). The starting position is 120 m higher than the end of the ramp, which is 3.00 m above the snow. Find the final speed of the skier if he lands 145 m down the 20.0° slope. Assume there is no friction on the ramp, but air resistance causes a 50% loss in the final kinetic energy. The GPS reading of the elevation of the skier is 4212 m at the top of the jump and 4039 m at the landing point.

To find the final speed of the skier, we can use the principle of conservation of energy. The total mechanical energy at the top of the ski ramp is equal to the total mechanical energy at the bottom of the slope, considering the loss in kinetic energy due to air resistance.

Let's break down the problem step by step:

1. Calculate the potential energy at the top and bottom of the ramp:
- Potential energy at the top (E₁) = mass × gravity × height
Here, the height is the difference in elevation between the starting position and the top of the jump: 120 m.
The GPS reading of the elevation at the top of the jump is given as 4212 m. So, the starting position is at 4212 m + 120 m = 4332 m.
- Potential energy at the bottom (E₂) = mass × gravity × height
Here, the height is the difference in elevation between the landing point and the bottom of the slope: 3 m.
The GPS reading of the elevation at the landing point is given as 4039 m. So, the bottom of the slope is at 4039 m.

2. Calculate the loss in potential energy due to elevation difference:
- Loss in potential energy (ΔPE) = E₁ - E₂

3. Calculate the loss in kinetic energy due to air resistance:
- Loss in kinetic energy (ΔKE) = 0.5 × initial kinetic energy

4. Calculate the final kinetic energy (KE):
- KE = initial kinetic energy - ΔKE

5. Finally, calculate the final speed (v) using the final kinetic energy:
- v = √(2 × KE / mass)

Note: In this explanation, we assume the mass of the skier is known. If not provided, it would need to be given or calculated using additional information.

Once all the necessary values are known, you can substitute them into the equations to calculate the final speed.

To find the final speed of the skier, we can break down the problem into multiple steps and use the laws of physics. Here are the step-by-step calculations:

Step 1: Find the initial potential energy of the skier:
The initial potential energy (PE) is given by the equation:
PE = m * g * h
Where:
m = mass of the skier (assuming it's 1 kg, which we'll use for simplicity)
g = acceleration due to gravity (9.8 m/s^2)
h = height difference between the starting position and the end of the ramp
= 120 m
= -120 m (negative sign to represent the downward direction)

PE = 1 kg * 9.8 m/s^2 * (-120 m)
PE = -1176 J (Joules)

Step 2: Find the final potential energy of the skier at the landing point:
The final potential energy (PE') is given by the equation:
PE' = m * g * h'
Where:
h' = height difference between the landing point and the end of the ramp
= 3 m
= -3 m (negative sign to represent the downward direction)

PE' = 1 kg * 9.8 m/s^2 * (-3 m)
PE' = -29.4 J (Joules)

Step 3: Calculate the change in potential energy:
ΔPE = PE' - PE
ΔPE = (-29.4 J) - (-1176 J)
ΔPE = 1146.6 J (Joules)

Step 4: Calculate the change in kinetic energy due to air resistance:
ΔKE = -0.5 * KE (50% loss in kinetic energy)
Where KE is the initial kinetic energy.

Step 5: Calculate the initial kinetic energy of the skier:
The initial kinetic energy (KE) is given by the equation:
KE = 0.5 * m * v^2
Where:
v = initial speed of the skier
= 4.00 m/s

KE = 0.5 * 1 kg * (4.00 m/s)^2
KE = 8 J (Joules)

Step 6: Calculate the change in kinetic energy:
ΔKE = -0.5 * KE
ΔKE = -0.5 * 8 J
ΔKE = -4 J (Joules)

Step 7: Calculate the total energy at the landing point:
The total energy (E') at the landing point is given by the equation:
E' = PE' + KE'
Since there is no friction on the ramp, the mechanical energy is conserved.

E' = PE' + KE' = ΔPE + ΔKE

E' = 1146.6 J + (-4 J)
E' = 1142.6 J (Joules)

Step 8: Find the final speed of the skier at the landing point:
The final speed (v') of the skier is given by the equation:
v' = √((2 * E') / m)
Where:
m = mass of the skier
= 1 kg

v' = √((2 * 1142.6 J) / 1 kg)
v' = √(2285.2 m^2/s^2) m/s
v' ≈ 47.835 m/s (approximately)

Therefore, the final speed of the skier at the landing point is approximately 47.835 m/s.