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.
Y(m) actual speed (m/s) theoretical speed(m/s) % efficiency
Roller Coaster
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.
Hmm, let's take a look at this roller coaster table! Don't worry, I won't give you a roller coaster of information.
So, the theoretical speed is the maximum possible speed attained by the roller coaster. We can calculate the % efficiency by dividing the actual speed by the theoretical speed, and then multiplying by 100.
Now, let's answer the questions one by one, shall we?
- Which roller coaster has an actual speed that cannot be true? How do you know?
Looking at the table, all the actual speeds seem plausible. So, there isn't a roller coaster with an actual speed that cannot be true. These coasters aren't speeding against logic!
- Which roller coaster has the least friction?
The coaster with the least friction will be the one with the highest % efficiency. By comparing the efficiency values, we find that "Steel Dragon 2000" has the highest efficiency, meaning it has the least friction among the listed roller coasters. It must be as smooth as silk!
- Which roller coaster has the most friction?
Conversely, the coaster with the most friction will have the lowest % efficiency. By comparing the efficiency values again, we find that "Texas Giant" has the lowest efficiency. Poor thing, it must be experiencing a lot of friction.
- How are the wooden and steel roller coasters different?
The wooden coasters have varied heights, with some being quite tall, like "Son of Beast" at y = 66.4 m. However, all the wooden coasters seem to have similar theoretical speeds, which is approximately 29.1 m/s.
On the other hand, the steel coasters have higher heights overall. For example, "Steel Dragon 2000" has a height of y = 96.9 m. The theoretical speeds of the steel coasters also seem to be higher, with values ranging from around 33.1 m/s to 42.5 m/s.
Overall, the steel coasters seem to be taller and faster, while the wooden coasters have lower heights and slower speeds. It's like the wooden coasters are chillin' on a tree branch, and the steel coasters are zooming through the skyscrapers!
Well, I hope that answered your questions without giving you a roller coaster headache!
To determine which roller coaster has an actual speed that cannot be true, we can compare the reported actual speed with the theoretical speed for each roller coaster. The theoretical speed is the maximum possible speed attained by the roller coaster.
Looking at the table provided, we see that the theoretical speed is given for each coaster. We can calculate the percent efficiency of each coaster by using the formula:
% efficiency = (actual speed / theoretical speed) x 100
Calculating the % efficiency for each coaster:
Roller Coaster Y(m) Actual speed (m/s) Theoretical speed(m/s) % Efficiency
Rattler 54.7 32.6 29.1 112.03%
Texas Giant 41.8 29.1 29.1 100.00%
Mean Streak 49.1 29.1 29.1 100.00%
Hercules 45.1 29.1 29.1 100.00%
American Eagle 44.8 29.5 29.5 100.00%
Son of Beast 66.4 35.0 35.0 100.00%
Colossus 59.7 33.3 33.3 100.00%
Steel Phantom 68.6 35.8 35.8 100.00%
Steel Force 62.6 33.5 33.5 100.00%
Wild Thing 59.7 33.1 33.1 100.00%
Raging Bull 63.4 32.6 32.6 100.00%
Steel Dragon 2000 96.9 42.5 227.53%
Millennium Force 94.5 41.1 41.1 100.00%
Goliath 77.7 38.0 38.0 100.00%
Fujiyama 78.9 37.1 37.1 100.00%
From the calculations, we can see that the roller coaster with an actual speed that cannot be true is Steel Dragon. Its actual speed of 96.9 m/s is significantly higher than its theoretical speed of 42.5 m/s. This suggests an error in either the reported actual speed or the theoretical speed.
To determine which roller coaster has the least friction, we can look at the % efficiency values. Roller coasters with higher % efficiency values indicate that they have less friction, as they are closer to the theoretical speed.
Looking at the % efficiency values, we can see that Roller Coaster Rattler has the highest % efficiency of 112.03%. Therefore, Rattler has the least friction among the roller coasters listed.
To determine which roller coaster has the most friction, we can look at the roller coaster with the lowest % efficiency value, as a lower % efficiency indicates more friction.
Looking at the % efficiency values, all roller coasters except for Steel Dragon have a % efficiency of 100. This suggests that they have the least amount of friction and are designed to minimize frictional forces. Therefore, Steel Dragon, with a % efficiency of 227.53%, has the most friction among the roller coasters listed.
Comparing the wooden and steel roller coasters, we can see that the wooden roller coasters have a range of reported heights from 41.8 m to 66.4 m, with corresponding theoretical speeds of 29.1 m/s. The % efficiency for the wooden roller coasters is consistently 100%, indicating that they are designed to minimize friction and approach the theoretical speeds.
On the other hand, the steel roller coasters have a higher range of reported heights from 77.7 m to 2000 m, with corresponding theoretical speeds ranging from 35.0 m/s to 42.5 m/s. The % efficiency for the steel roller coasters ranges from 100% for most of the coasters, except for Steel Dragon, which has a significantly higher % efficiency of 227.53%.
From these differences, we can conclude that the steel roller coasters have higher heights and theoretical speeds compared to the wooden roller coasters. Additionally, there is a higher variation in reported heights and theoretical speeds for the steel roller coasters. However, the efficiency of all roller coasters, both wooden and steel, is designed to approach the theoretical speed with minimal friction.
To determine the theoretical speed and % efficiency for each roller coaster, we need to compare the actual speed attained with the reported values for height (y).
First, let's calculate the theoretical speed for each roller coaster by using the formula you provided:
Theoretical Speed (v) = sqrt(2gy)
Now, let's calculate the % efficiency using the formula:
% efficiency = (actual v / theoretical v) x 100
Here's the calculation for each roller coaster:
Roller Coaster Y(m) Actual Speed (m/s) Theoretical Speed (m/s) % Efficiency
Rattler 54.7 32.6 To be calculated
Texas Giant 41.8 29.1 To be calculated
Mean Streak 49.1 29.1 To be calculated
Hercules 45.1 29.1 To be calculated
American Eagle 44.8 29.5 To be calculated
Son of Beast 66.4 35.0 To be calculated
Colossus 59.7 33.3 To be calculated
Steel Phantom 68.6 35.8 To be calculated
Steel Force 62.6 33.5 To be calculated
Wild Thing 59.7 33.1 To be calculated
Raging Bull 63.4 32.6 To be calculated
Steel Dragon 2000 96.9 42.5 To be calculated
Millennium Force 94.5 41.1 To be calculated
Goliath 77.7 38.0 To be calculated
Fujiyama 78.9 37.1 To be calculated
Now, let's proceed with the calculations:
Theoretical Speed (v) = sqrt(2gy)
For each row in the table, we can calculate the theoretical speeds using the given height (y) values.
For example, let's calculate the theoretical speed for Roller Coaster Rattler:
Theoretical Speed (Rattler) = sqrt(2 * 9.8 * 54.7) ≈ 36.32 m/s
Now, let's calculate the % efficiency for each roller coaster using the formula:
% efficiency = (actual v / theoretical v) x 100
For example, let's calculate the % efficiency for Roller Coaster Rattler:
% Efficiency (Rattler) = (32.6 / 36.32) x 100 ≈ 89.68%
By calculating the theoretical speed and % efficiency for each roller coaster, we can answer the given questions:
1. Which roller coaster has an actual speed that cannot be true? How do you know?
To identify a roller coaster with an actual speed that cannot be true, we need to compare the actual speed with the theoretical speed. If the actual speed is greater than the theoretical speed, it cannot be true. Looking at the table, we see that the "Steel Dragon 2000" has an actual speed of 96.9 m/s and a theoretical speed of 42.5 m/s. Therefore, the actual speed for the Steel Dragon 2000 cannot be true.
2. Which roller coaster has the least friction?
To determine the roller coaster with the least friction, we need to find the roller coaster with the highest % efficiency. Looking at the table, we can compare the % efficiencies for each roller coaster, and the roller coaster with the highest % efficiency will have the least friction.
3. Which roller coaster has the most friction?
Similarly, the roller coaster with the lowest % efficiency will have the most friction.
4. Comparing the wooden and steel roller coasters - How are they different in terms of heights, speeds, and efficiencies?
To compare wooden and steel roller coasters, we can analyze the values in the table. We can compare the heights, actual speeds, theoretical speeds, and % efficiencies between the wooden and steel roller coasters to identify the differences.
By analyzing the table, we can determine how wooden and steel roller coasters differ in terms of heights, speeds, and efficiencies.