A ski slope drops at an angle of 24 degrees with respect to the horizontal and is 500 m long. A 60 kg skier skis down from the top to the bottom of the slope.
A.) Determine the change in gravitational potential energy of the skier+earth system.
b.) If 20% of the gravitation potential energy of the system is transformed into the kinetic energy of the skier, how fast is the skier going at the bottom of the slope?
A.) the height of the slope is
___ 500m * sin(24º)
GPE = m g h
b.) .2 m g h = 1/2 m v^2
.4 g h = v^2
To determine the change in gravitational potential energy of the skier+earth system, we can use the formula:
∆PE = mgh
Where ∆PE is the change in gravitational potential energy, m is the mass of the skier (60 kg), g is the acceleration due to gravity (approximately 9.8 m/s²), and h is the vertical height.
To find the vertical height, we can use trigonometry. The ski slope drops at an angle of 24 degrees with respect to the horizontal and is 500 m long. The vertical height (h) can be found using the equation:
h = length * sin(angle)
h = 500 * sin(24°)
Using a scientific calculator, we find that h ≈ 206.2 m.
Now we can calculate the change in gravitational potential energy:
∆PE = (60 kg) * (9.8 m/s²) * (206.2 m)
∆PE ≈ 120,471.6 J
Therefore, the change in gravitational potential energy of the skier+earth system is approximately 120,471.6 Joules.
b.) If 20% of the gravitational potential energy is transformed into kinetic energy, we can calculate the initial potential energy (PE_initial) and the final kinetic energy (KE_final) using the following formulas:
PE_initial = 0.2 * ∆PE
KE_final = 0.8 * ∆PE
Substituting the value of ∆PE (120,471.6 J), we get:
PE_initial = 0.2 * 120,471.6 J
PE_initial ≈ 24,094.32 J
KE_final = 0.8 * 120,471.6 J
KE_final ≈ 96,377.28 J
The final kinetic energy (KE_final) is equal to the initial potential energy (PE_initial). We can use the formula for kinetic energy:
KE_final = (1/2) * m * v²
Rearranging the equation to solve for velocity (v), we get:
v = sqrt((2 * KE_final) / m)
Substituting the values, we get:
v = sqrt((2 * 96,377.28 J) / 60 kg)
Using a calculator, we find that v ≈ 19.85 m/s.
Therefore, the skier is going approximately 19.85 m/s at the bottom of the slope.