Analyzing Roller Coaster Performance
Using Conservation of Mechanical Energy
At the beginning of a roller coaster ride, the car is lifted to the top of a large hill and
released. The speed of the car at the top of the hill is small, so we will assume it to be
zero. The car rolls freely down this hill and reaches its maximum speed at the bottom.
If the roller coaster were frictionless, mechanical energy would be conserved… Ei = Ef.
Showing all terms for potential and kinetic energy, set up the conservation of mechanical
energy for this situation…
yi
vf
Solve this relationship for the maximum speed of the car, vf, in terms of height, yi .
This speed is the maximum possible speed attained by the roller coaster, or the
“theoretical speed”. Roller coasters are carefully designed to minimize frictional forces
so that they approach these theoretical speeds. The efficiency of a roller coaster design
can be found by comparing the actual speed attained with this theoretical speed...
% efficiency = (actual v / theoretical v ) x 100
The table below gives reported values for height (y) and maximum speed attained by
actual roller coasters. Determine the theoretical speed and the % efficiency for each
coaster.
Roller Coaster y
(m)
actual speed
(m/s)
theoretical speed
(m/s)
% efficiency
Rattler 54.7 32.6
Texas Giant 41.8 29.1
Mean Streak 49.1 29.1
Hercules 45.1 29.1
American Eagle 44.8 29.5
Son of Beast 66.4 35.0
Colossus 59.7 33.3
Steel Phantom 68.6 35.8
Steel Force 62.6 33.5
Wild Thing 59.7 33.1
Raging Bull 63.4 32.6
Steel Dragon 2000 96.9 42.5
Millennium Force 94.5 41.1
Goliath 77.7 38.0
Fujiyama 78.9 37.1
Which roller coaster has an actual speed that cannot be true? How do you know?
Which roller coaster has the least friction?
Which roller coaster has the most friction?
The first seven roller coasters listed are made of wood and the last eight are made of
steel. Use your table to compare the wooden and steel roller coasters. How are they
different? Consider heights, speeds, and efficiencies.
1/2 m vf^2=mgh
vf=sqrt2gh
To solve this problem, we will use the conservation of mechanical energy. According to this principle, the initial mechanical energy (potential energy) will be equal to the final mechanical energy (kinetic energy) if there is no friction involved.
The initial mechanical energy at the top of the hill (Ei) is equal to the potential energy (PE) and can be expressed as:
Ei = m * g * y
The final mechanical energy at the bottom of the hill (Ef) is equal to the kinetic energy (KE) and can be expressed as:
Ef = (1/2) * m * vf^2
By setting up the conservation of mechanical energy equation, we can solve for the maximum speed of the car, vf, in terms of height, yi:
m * g * y = (1/2) * m * vf^2
Here, m represents the mass of the car, g represents the acceleration due to gravity, y represents the height of the hill, and vf represents the maximum speed of the car.
Simplifying the equation by canceling out the mass (m) term:
g * y = (1/2) * vf^2
To solve for vf, we can rearrange the equation:
vf^2 = 2 * g * y
Finally, taking the square root of both sides of the equation:
vf = √(2 * g * y)
Now that we have the equation to calculate the maximum speed of the car, we can proceed to answer the questions.
To determine which roller coaster has an actual speed that cannot be true, we can compare the maximum speeds reported in the table to the calculated theoretical speeds using the equation vf = √(2 * g * y). If any reported speed is greater than the corresponding theoretical speed, it cannot be true.
To determine which roller coaster has the least friction, we can look for the roller coaster with the highest percentage efficiency. The roller coaster with the highest percentage efficiency indicates minimal energy loss due to friction.
To determine which roller coaster has the most friction, we can look for the roller coaster with the lowest percentage efficiency. The roller coaster with the lowest percentage efficiency indicates significant energy loss due to friction.
To compare the wooden and steel roller coasters, we can analyze the heights, speeds, and efficiencies. We can look for any patterns or differences between the two types of roller coasters in terms of these parameters.