Engineering models 1

Lab 5: There (Over a Hill) and Back Again: A Roller Coaster Physics Tale!

I. Background

At a fundamental level, a roller coaster operates due to one simple physics concept: conservation of energy. As a roller coaster reaches the top of the first hill, it must have all of the energy it needs in order to make it through all of the various hills, loops, twists, and turns and return to the loading area (unless it gets a boost at some point). At this highest point, all (or most) of the energy of the roller coaster is stored as potential energy. As the roller coaster travels through its track, that energy is converted between potential energy and kinetic energy. In the real world, additional forces (air resistance, friction, braking at the end) act on the roller coaster and cause some of the original potential energy to be lost as work. For the purposes of this lab, we will ignore the effects of these additional forces and look solely at the potential and kinetic energy of the roller coaster.

The roller coaster we will be analyzing is depicted below and consists of a single initial hill followed by a single loop.

r

h

L

θ

In the diagram, r is the radius of the loop, h is the height of the initial hill, L is the length of the first hill, and θ is the angle of the first hill. The equations shown above describe the energy of the system. The total energy is equal to the sum of the potential and kinetic energy of the system. For the kinetic energy, m is the mass of the roller coaster (in kg) and v is the velocity of the roller coaster (in m/s). For the potential energy, m is again the mass of the roller coaster, g is the acceleration due to gravity (9.81 m/s2), and h is the height of the roller coaster (in m).

As the roller coaster moves down the initial hill, all of its original potential energy is converted into kinetic energy. It then uses this kinetic energy to make it through the loop. Therefore, it must have enough kinetic energy at the beginning of the loop to make it to the top of the loop and still have kinetic energy left to continue moving through the peak and down the other side.

If we focus only on the initial hill, we can determine the amount of time it takes to get from the top of the hill to the bottom and the velocity and height of the roller coaster during its travels down the hill.

If we draw the free body diagram of the roller coaster on the hill, we can find the acceleration of the roller coaster. Using this, we can derive an expression for time at which the position of the roller coaster is 0 (i.e. it has reached the bottom of the hill), the height of the roller coaster, and the velocity of the roller coaster:

m*g m*g*sin(θ) m*g*cos(θ) θ L θ h

Another consideration is the amount of force the rider feels, often referred to as the g-force. Excessive force can cause riders to become nauseated or black out, which reduces the enjoyment of the ride. The maximum g-force occurs at the bottom of the loop after coming down the initial hill. The force experienced by the rider is the sum of the force due to gravity (in the downward direction) and the normal force exerted by the car and the track (in the upward direction). We know from Newton’s second law that the force F on an object is equal to the mass of the object times its acceleration, or F = ma. Using this and the equation for the total force experienced by the user below, the net acceleration of the rider can be determined.

The g-force experienced by an object is not an actual force, but a rather how many times the acceleration due to gravity an object experiences. The g-force can be determined using the equation below.

In the remainder of this lab, you will use these physics ideas to develop an Excel spreadsheet to help you analyze different roller coaster configurations to determine if they are both safe and functional.

II. Practice

For this part of the lab, download the Lab_05.xlsx workbook and locate the Part II worksheet. Complete this worksheet by doing the following (for any calculations, use g = 9.81 m/s2):

Energy:

· In cells F4:F6, compute the potential, kinetic, and total energy based on user specified values for the mass, height and velocity in cells C4, C5, and C6, respectively

· You should use conditional statements so that

· the potential energy is not calculated unless there are values for the mass and the height

· the kinetic energy is not calculated unless there are values for the mass and the velocity

· the total energy is not calculated unless there are values for both the potential and kinetic energy

Test your worksheet with the following sets of inputs:

Mass	Height	Velocity	Potential Energy	Kinetic Energy	Total Energy
100	50	0
100	0	10
100	50	10

G-Force:

· In cell F10, calculate the G-Force experienced by a person based on the mass, velocity, and radius of the circular path specified by the user in cells C11 and C12, respectively

· You should use conditional statements so that the G-Force is not calculated until all three values have been entered

· Add a conditional statement to cell F11 to determine if the situation is safe for a person, where the average person can experience at most 5g’s before blacking out

· No message should be displayed if no value is displayed in cell F10

Test your worksheet with the following sets of inputs:

Velocity	Radius	G-Force	Safe?
10	5
20	5
10	1

III. Roller Coaster Time!

Now that you’ve had the opportunity to perform some basic calculations and conditionals, it’s time to develop the worksheet that will analyze the performance of our roller coaster. Locate the Part III worksheet in the Lab_05.xlsx workbook and follow the steps below to complete the worksheet (again, use g = 9.81 m/s2):

Roller Coaster Parameters:

· In cell D7, create an Excel statement that will compute the total mass of the roller coaster; it should only display the results of the calculation if values are specified for all four parameters (number of riders, average mass of a rider, number of cars, mass of a car)

Results:

· For each of the calculations below, a value should only be displayed if the total mass and the values for the track parameters have been entered

· In cell D16, calculate the total energy (potential + kinetic) at the top of the hill

· In cell D17, calculate the amount of time it will take the roller coaster to reach the bottom of the first hill using the equation shown above

· In cell D18, calculate the velocity of the roller coaster at the bottom of the hill; this can be done either by determining the increase in velocity due to the conversion of potential into kinetic energy or by plugging in your value for into the equation for

· In cell D19, calculate the maximum G-Force experienced by the riders by using the formula for G-Force above and the value you calculated for the velocity at the bottom of the hill

· In cell D20, calculate the potential energy at the top of the loop

· In cell D21, determine whether there is any kinetic energy at the top of the loop by finding the difference between the total energy in cell D16 and the potential energy in cell D20; if the difference is not positive you should display nothing and if the difference is positive, display the kinetic energy (which is the difference)

· In cell D22, calculate the velocity of the roller coaster at the top of the loop using the kinetic energy calculated in cell D21 as long as the kinetic energy was able to be calculated; if the kinetic energy could not be calculated, display nothing

· In cell D23, determine if the current configuration for the roller coaster would be functional (i.e. the roller coaster is able to make it through the loop)

· Add conditional formatting for cell D23 so that if the configuration is functional, the background is green and if the configuration is not functional, the background is red

· In cell D24, determine if the current configuration is safe for the riders (i.e. they will not experience excessive G-Force)

· Add conditional formatting for cell D24 so that if the configuration is safe, the background is green and if the configuration is not safe, the background is red

Performance on Hill:

· In cells F5:F23, use Excel formulas to create a set of 20 evenly spaced time values ranging from 0 to your calculated value for the length of time to make it down the hill

· In cells G4:G23, use Excel formulas to calculate the distance the roller coaster has moved down the track from its initial position at the top of the hill at the time values in column F

· In cells H4:H23, use Excel formulas to calculate the height of the roller coaster as it moves down the hill at the time values in column F

· In cells I4:I23, use Excel formulas to calculate the velocity of the roller coaster as it moves down the hill at the time values in column F

· In cells J4:J23, use Excel formulas to calculate the potential energy of the roller coaster as it moves down the hill at the time values in column F

· In cells K4:K23, use Excel formulas to calculate the kinetic energy of the roller coaster as it moves down the hill at the time values in column F

· In cells L4:L23, use Excel formulas to calculate the total energy of the roller coaster as it moves down the hill at the time values in column F

Energy Graph:

· Create a scatter plot with points connected by lines that shows both the potential and the kinetic energy of the roller coaster as it moves down the hill (with time on the x-axis) and place it in the location indicated on the worksheet

Once you have your worksheet complete, use it to evaluate the configurations below.

Configure 1:

Number of Riders:	10		Total Energy:
Avg. Mass of Rider:	100 kg		Time for Hill:
Number of Cars:	2		Velocity at Bottom:
Mass of Car:	1500 kg		Max G-Force
Initial Height:	50 m		Pot. Eng. at Loop Peak:
Angle of Hill:	45o		Kin. Eng. at Loop Peak:
Initial Velocity:	10 m/s		Velocity at Loop Peak:
Radius of Loop:	20 m		Functional?
			Safe?

Paste Energy Graph Below:

Configure 2:

Number of Riders:	10		Total Energy:
Avg. Mass of Rider:	100 kg		Time for Hill:
Number of Cars:	2		Velocity at Bottom:
Mass of Car:	1500 kg		Max G-Force
Initial Height:	50 m		Pot. Eng. at Loop Peak:
Angle of Hill:	45o		Kin. Eng. at Loop Peak:
Initial Velocity:	10 m/s		Velocity at Loop Peak:
Radius of Loop:	30 m		Functional?
			Safe?

Paste Energy Graph Below:

Configure 3:

Number of Riders:	10		Total Energy:
Avg. Mass of Rider:	100 kg		Time for Hill:
Number of Cars:	2		Velocity at Bottom:
Mass of Car:	1500 kg		Max G-Force
Initial Height:	50 m		Pot. Eng. at Loop Peak:
Angle of Hill:	45o		Kin. Eng. at Loop Peak:
Initial Velocity:	20 m/s		Velocity at Loop Peak:
Radius of Loop:	20 m		Functional?
			Safe?

Paste Energy Graph Below:

Submit this completed document and your Excel file to your section site on Blackboard.