To solve this problem, we can use the principles of conservation of mechanical energy and apply the work-energy theorem.
First, we need to find the potential energy of the roller coaster at the top of the hill. The potential energy can be calculated using the formula:
PE = m * g * h
Where:
PE is the potential energy
m is the mass of the roller coaster (230 kg)
g is the acceleration due to gravity (9.8 m/s^2)
h is the height of the hill
Since the roller coaster reaches the top of the hill, its height h is equal to the vertical displacement caused by climbing the hill. We can calculate this using trigonometry:
h = d * sin(θ)
Where:
d is the length of the hill (50.0 m)
θ is the angle of the hill (25°)
Now, we can substitute the values into the formulas:
h = 50.0 m * sin(25°)
h ≈ 21.18 m
PE = 230 kg * 9.8 m/s^2 * 21.18 m
PE ≈ 46855 J
Next, we need to find the kinetic energy of the roller coaster at the bottom of the hill. Since the roller coaster loses some energy due to friction, we need to take into account the coefficient of kinetic friction (µk).
The work-energy theorem states that the net work done on an object is equal to the change in its kinetic energy:
Work = ∆KE
The net work done is equal to the sum of the work done by gravity and the work done by friction:
Work = Work_gravity + Work_friction
Work_gravity = PE (potential energy at the top of the hill)
Work_friction = µk * m * g * d (friction force * distance)
Now, let's calculate the work done by friction:
Work_friction = 0.16 * 230 kg * 9.8 m/s^2 * 50.0 m
Work_friction ≈ 17984 J
Substituting the values into the equation:
Work = PE + Work_friction
∆KE = PE + Work_friction
∆KE = 46855 J + 17984 J
∆KE ≈ 64839 J
Therefore, the kinetic energy of the roller coaster when it reaches the bottom of the hill is approximately 64839 J.