A roller coaster reaches the top of the steepest hill with a speed of 3.0 km/h . It then descends the hill, which is at an average angle of 40 degrees and is 50.0 m long. Estimate its speed when it reaches the bottom. Assume kinetic friction (uk) = 0.18.
To estimate the speed of the roller coaster when it reaches the bottom of the hill, we can use the principles of energy conservation and analyze the forces acting on the roller coaster.
First, let's convert the initial speed of the roller coaster at the top of the hill from kilometers per hour to meters per second:
3.0 km/h = 3.0 * 1000 / 3600 = 0.833 m/s
Next, let's calculate the potential energy (PE) and kinetic energy (KE) at the top of the hill. The roller coaster's potential energy at the top can be calculated using the equation:
PE = m * g * h
Where:
m = mass of the roller coaster
g = acceleration due to gravity
h = height of the hill
Since the roller coaster reaches the top of the hill with a certain speed, it also has kinetic energy, which can be calculated using the equation:
KE = (1/2) * m * v^2
Where:
v = velocity of the roller coaster
Since energy is conserved, the total energy at the top of the hill (TE) is equal to the sum of the potential and kinetic energies:
TE = PE + KE
At the bottom of the hill, the roller coaster will have a final speed. We can find this speed by using the concept of mechanical energy conservation, neglecting losses due to friction. The total energy at the bottom (TE') is equal to the sum of the potential and kinetic energies:
TE' = PE' + KE'
However, at the bottom of the hill, there is friction acting on the roller coaster. The work done by friction (W_f) can be calculated using the equation:
W_f = uk * m * g * d
Where:
uk = coefficient of kinetic friction
d = distance traveled
To find the final potential energy (PE') at the bottom, we need to take into account the difference in height between the top and bottom of the hill. In this case, the roller coaster goes down the hill, so the change in height (Δh) is negative:
PE' = m * g * (h - Δh)
Now we can solve for the final kinetic energy (KE') at the bottom:
KE' = TE' - PE'
Substituting the equations and values we have, we can solve for the final velocity (v'):
TE' = PE' + KE'
TE' = m * g * (h - Δh) + KE'
v' = sqrt(2 * (TE' - m * g * (h - Δh)) / m)
Given the following values:
Angle of the hill = 40 degrees
Length of the hill = 50.0 m
Coefficient of kinetic friction (uk) = 0.18
Initial velocity (v) = 0.833 m/s
We can now calculate the final velocity (v') at the bottom of the hill.