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.

Question

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.

Answer 1

To find the final speed of the skier, we can use the principle of conservation of mechanical energy, which states that the total mechanical energy of a system remains constant if no external work is done on it. In this case, the system consists of the skier.

The mechanical energy of the skier is given by the sum of its potential energy and kinetic energy:

E = PE + KE

1. Calculate the potential energy (PE) at the starting position:
PE_start = m * g * h_start

Given:
- m (mass of the skier) is not provided in the question.
- g (acceleration due to gravity) is approximately 9.8 m/s².
- h_start (height at the starting position) is the elevation at the top of the jump reading minus the elevation at the landing point reading.

h_start = 4212 m - 4039 m = 173 m

PE_start = m * 9.8 m/s² * 173 m

2. Calculate the potential energy (PE) at the end of the ramp:
PE_end = m * g * h_end

Given:
- h_end (height at the end of the ramp) is 3.00 m.

PE_end = m * 9.8 m/s² * 3 m

3. Calculate the kinetic energy (KE) at the end of the ramp:
KE_end = (1/2) * m * v_end^2

Given:
- v_end (final speed at the end of the ramp) is what we need to find.

4. Calculate the final kinetic energy (KE_final) after accounting for the 50% loss due to air resistance:
KE_final = KE_end * (1 - 0.50)

5. Calculate the kinetic energy (KE) at the landing point using the principle of conservation of mechanical energy:
KE_landing = KE_start - (PE_end - PE_start) - KE_final

Given:
- KE_start is equal to the initial kinetic energy and is not given in the question.

6. Calculate the final speed (v_landing) at the landing point:
v_landing = sqrt((2 * KE_landing) / m)

Now we have all the information to find the final speed of the skier. Please provide the missing value for the mass of the skier, and we can proceed with the calculation.