Rollercoaster Rules

(Download the MS Word version of this document.)
(Download the MS Excel version of the rollercoaster spreadsheet.)

Adapted from Active Physics (AAPT). It's About Time, Herff Jones Education Division

Your lab group will design a rollercoaster with the following specifications:

I. Materials:

• 1 ft x 2.5 ft piece of cardboard (provided by the student)
• masking tape (provided by the student)
• 1.1 meters of bendable wire, representing 1000 m of track (The first 0.1 meters can be used for the first uphill climb.)
• Hot glue and gun for attaching the wire to the cardboard
• decorations (provided by the student)

II. Your rollercoaster will include:

• a rollercoaster car of mass 4500 kg
• a starting hill 60 meters high (use a scale of 1 cm = 10 m)
• a vertical circular loop
• a second hill
• a horizontal circular loop
• enough speed left at the end to roll back into the starting station

III. The total energy lost throughout the 1000 meters of track is 2/3 of the original energy. Calculate the energy lost per meter of track so that you will know how much energy the car is losing for each scaled meter of track as you design the rollercoaster.

IV. In the data table on the back, show your calculations for height, potential energy, kinetic energy, and velocity at the points on your rollercoaster shown below:

V. After you find the velocity at point F, calculate the centripetal force acting on the car immediately after it enters the horizontal loop.

VI. After you find the velocity at point G, use kinematics to show that the car can stop by the time it covers the distance between points G and H with an acceleration which is no larger than -10 m/s².

Rollercoaster Example Problems

1. There is 100 meters of track between points A and B. A roller coaster of mass 4500 kg starts from rest at point A. One third of the total energy is lost by the time the car reaches point B. If the speed of the car is to be 20 m/s,

(a) what is the total energy lost?

(b) what is the total energy lost per meter of track?

(c) what must be the height at point B?

2. What if we wanted to include a vertical loop in our roller coaster?

We would have to review what we know about circular motion.

Are you lighter at the top of the loop or the bottom? ____________

Draw forces at the top and bottom of the RC loop:

3. Horizontal loop:

(a) How much force does a horizontal circular track exert on a 4500 kg rollercoaster car moving at an average speed of 14 m/s, if the radius of the loop is 40 m?

(b) What is the centripetal acceleration of the car?

(c) How many "g's" is this acceleration?

Another Rollercoaster Example Problem

GIVEN: The mass of the cart is 4000 kg, the length of the track is 800 m, and 3/4 of the total energy is lost to heat and friction over the entire track.

1. What is the total energy of the rollercoaster? What is the potential energy at point A?

(These two questions are the same)

2. What is the energy lost per meter of track?

3. If there is 80 m of track from point A to point B, what is the kinetic energy at point B?

4. What is the kinetic energy at point C?

5. What is the speed of the car at point C?

6. What is the centripetal acceleration at point C? How many g's is this?

7. What is the kinetic energy at point D at the bottom of the loop?

8. What is the potential energy at point E?

9. If the distance between points D and E is 60 m, what is the kinetic energy at point E?

10. If the kinetic energy at point F is 600,000 J, what is the kinetic energy at point G as the car exits the loop?

11. Using the kinetic energy you calculated for point G,

(a) what is the velocity at point G?

(b) what would the distance between points G and H have to be for the coaster to decelerate at - 5 m/s² to stop at point H?

Getting Started on Your Rollercoaster Model

1. Obtain a 1.1 meter wire from your teacher. Using a permanent marker, make a mark every 2 cm on the wire. You may also want to make a tape measure out of masking tape.

2. Bend the wire in the shape of the rollercoaster to include a 60-m (6cm) lift hill, a vertical loop, a second hill, a horizontal loop, and a stopping distance. Be sure the height of the lift hill is 6 cm, and the other elements are shorter than that. At this point you are just estimating the sizes of the vertical loop and the second hill.

3. Draw a straight horizontal line on your bookcover, and tape your wire flat to the bookcover, with the bottom of the wire on the horizontal line. Begin measuring all heights (from the horizontal line) and lengths along the track from point A to points B through H.

4. Begin filling in your table with the values you measure at each point. Remember the mass of your rollercoaster car is 4500 kg, and the height of the first hill is 60 meters. The total energy lost is 2/3 of the total energy (potential energy at A), so you can find the energy lost per meter of track.

5. When you finish filling in all of your table, you are ready to put your numbers into the spreadsheet.

1. Microsoft Applications
2. MS Excel 2000
3. File - Open
4. Template Drive
5. All Physics
6. Physics I
7. Phy I Rollercoaster Calculations.xls
8. Click on each tab at the bottom to enter data for each part of the rollercoaster.

Equations

	PE1	KE1		EL(1-2)		PE2	KE2
Point / Interval	Potential Energy (J)	Kinetic Energy (J)	Distance between points (m)	Energy Lost (J)	Height at end of interval (m)	Potential Energy Remaining (J)	Kinetic Energy Remaining (J)	Velocity at end point (m/s)
A		0	n/a	0	60		0	0
A→B
B→C								ac = g's =
C→D
D→E
E→F
F→G								ac = g's =
G→H
G→H	a* =				0	0	0	0

PE₁ + KE₁ - E_L(1-2) = PE₂ + KE₂

* Acceleration required to stop car between G and H:
W = KE_2(G-H) + E_L(G-H) = mad_G-H