Physics Roller Coaster Project Report: To design and construct a roller coaster utilizing the Princi

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Physics Roller Coaster Project Report Eknoor Singh & Vedant Shah Mr. Wu 12/23/2017

2 Table of Contents Introduction3 Materials 3 Drawings 4-5 Calculations6-10 Analysis 10-12 References 13

3 Introduction The purpose of this physics culminating project was to design and construct a roller coaster utilizing the Principle of Conservation of Energy. The Principle of Conservation of Energy is a principle that states that energy cannot be destroyed nor can it be created. It also states that the total energy inside a closed/isolated system remains the same (Nave). This principle was discovered by Julius Robert Mayer in 1842. The roller coaster was required to have be self-supporting, weigh less than 3.0 pounds, have a jump were it leaves the track at an angle of 30° and land on a new section of the track at a different level from which it left the jump at, have at least one loop, have at least one hill, have a maximum volume of 0.60 m3 with a maximum length of 0.70 m in any given direction, and fit a marble “car”. Additionally, it also had to have all routes open and visible. Lastly, this project had to incorporate physics principles that was taught in the course.

Materials ● Marble (4.2 gram), Quantity: 1 ● Polyurethane Tubing, Quantity: 25 feet ● Styrofoam Boards (14½ inches x 48 inches x 1½ inches), Quantity: 5 ● Sponges, Quantity: 2 ● Dowels (⅜ inch x 48 inches), Quantity: 6 ● Popsicle Sticks, Quantity: 75 ● Red Acrylic Paint (250ml), Quantity: 2 tubes ● White Acrylic Paint (250ml), Quantity: 1 tube ● Project Panels (60cm x 60cm x 2.5cm), Quantity: 2 ● Hot Glue Gun, Quantity: 1 ● Glue Sticks, Quantity: 2

4 Diagrams

Figure 1: The front view of the roller coaster. (1) The maximum height of the roller coaster is 85cm, where it has the most gravitational potential energy. (2) Here, the marble faces its first hill, where it gains some gravitational potential energy due to the loss of kinetic energy as there is an increase in height, causing the marble to lose some velocity. (3) After, it experiences the loop where more gravitational potential energy is converted to kinetic energy. (4) Now, the marble faces it second hill, where it loses some of its kinetic energy which is transferred to gravitational potential energy as it goes up the hill. (5) When the marble goes through the jump, it gains some gravitational potential energy while going up and loses kinetic energy, but when it is going back towards the track, it loses gravitational potential energy and gains kinetic energy. (6) At this point, all the remaining gravitational potential energy is converted into kinetic energy. Therefore, the gravitational potential energy at (1) when the marble is at rest is equal to the kinetic energy at (6) due to the Law of Conservation of Energy as air resistance and friction is being ignored.

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Figure 2: The side view of the roller coaster. (1) The maximum height of the roller coaster where the marble is dropped to at a height of 85 cm, where it has the most gravitational potential energy. (2) Here, the marble faces its first hill, where it gains some gravitational potential energy due to the loss of kinetic energy as there is an increase in height, causing the marble to lose some velocity. (3) After, it experiences the loop where more gravitational potential energy is converted to kinetic energy. (4) Now, the marble faces it second hill, where it loses some of its kinetic energy which is transferred to gravitational potential energy as it goes up the hill. (5) When the marble goes through the jump, it gains some gravitational potential energy while going up and loses kinetic energy, but when it is going back towards the track, it loses gravitational potential energy and gains kinetic energy. (6) At this point, all the remaining gravitational potential energy is converted into kinetic energy. Therefore, the gravitational potential energy at (1) when the marble is at rest is equal to the kinetic energy at (6) due to the Law of Conservation of Energy as air resistance and friction is being ignored.

6 Calculations *Note: All calculations are done neglecting friction on the track and air resistance.

Mass of marble=0.0042 kg m ⃗g=9.8 2 [down] s a. Gravitational energy at the top of the first drop Equation used to determine the value of gravitational energy at the top of the first drop

E g=mgh

Calculations

E g=mgh E g=(0.0042)(9.8)(0.74 ) E g ≅ 0.030458

E g ≅ 0.030 J

Therefore,

E g ≅ 0.030 J

at the top of the first drop.

b. Kinetic energy at the bottom of the first drop (using Conservation of Mechanical Energy) Equation used to determine the value of kinetic energy at the bottom of the first drop

ET @ Topofrollercoaster =ET @ Bottomoffirstdrop E g+ E k =E g + Ek

1 mgh+ mv 2 =mgh+ E k 2 Calculations

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1 mgh+ mv 2 =mgh+ E k 2 1 (0.0042)(9.8)(0.85)+ (0.0042)(0)2 ≅ 0.030458+ Ek 2 0.034986+0 ≅ 0.030458+ Ek 0.034986−0.030458≅ Ek

0.0045280 ≅ E k 0.0045 J ≅ Ek

Therefore,

Ek ≅ 0.0045 J

at the bottom of the first drop.

c. Velocity of the ball at the bottom of the first drop Equation used to determine the value of velocity of the marble at the bottom of the first drop

1 Ek = mv 2 2 Calculations

1 Ek = mv 2 2 1 2 0.0045280 ≅ ( 0.0042) v 2 1 0.0045280 ≅ ( 0.0042) v 2 2 0.0045280 0.0021 2 ≅ v 0.0021 0.0021 √❑

1.46840 m/s ≅ v

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1.5 m/s ≅ v Therefore, v ≅1.5 m/ s

of the marble at the bottom of the first drop.

d. i. Velocity of the ball as it leaves the 30 degree ramp Let Eg1 represent the gravitational energy present at the start of the track.

Let Eg2 represent the gravitational energy present at the top of the ramp.

Let Eg1 represent the kinetic energy present at the top of the ramp.

Equation used to determine the value of velocity of the marble at the bottom of the first drop

ET =ET '

E g1 =Eg 2 + E k2 Calculations

1 2 m ⃗g h1 =m ⃗g h2 + m ⃗ v 2 1 2 ⃗g h1−⃗g h2= ⃗ v 2 √❑ √❑

3.6373 m/s ≅ ⃗v 3.6 m/s ≅ ⃗v Therefore the velocity of the ball leaving the ramp is 3.6 m/s. ii. Duration of the Flight (Projectile Motion) Let [up] be positive.

Let

vsin 30 represent the vertical component of initial velocity

Equation used to determine the value of velocity of the marble at the bottom of the first drop

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1 ∆ d⃗ =⃗v i ∆ t + a ⃗ ∆ t2 2 Calculations −0.1=⃗v sin 30 ∆ t−4.9 ∆ t

2

0=−4.9 ∆ t 2 +1.81865 ∆ t +¿ 0.1 t=

−b ± √❑ ❑

t=

−1.81865 ± √ ❑ ❑

t=

−1.81865 ± √❑ ❑

Reject +. We are looking for positive time.

t=0.46 s

iii. Horizontal Distance Covered (Projectile Motion) and % Error Let

vcos 30 represent the horizontal component of initial velocity

Equation used to determine the value of velocity of the marble at the bottom of the first drop

∆ d⃗ =⃗ vi ∆ t Calculations

∆ d⃗ =⃗v cos 30∗0.46 ∆ d⃗ =3.14999∗0.46

∆ d⃗ =1.44899 Actual Value = 0.14m, Theoretical Value = 1.44899

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Error=¿

Theoretical− Actual ∨× 100 Theoretical

Error=¿

1.44899−0.14 ∨×100 1.44899 Error=¿ 90.3%

iv. Maximum Height of Projectile and Time of Maximum Height Let

vsin 30 represent the vertical component of the initial velocity.

Equation used to determine the value of velocity of the marble at the bottom of the first drop

⃗ V f 2 =⃗ V i2+ 2 a⃗ h max 2

0=⃗v sin 30 + 2(−9.8) hmax

0=1.818652+ 2(−9.8)hmax hmax =

3.307487 19.6

hmax =0.17 m

Time at Max Height

⃗ V f =⃗ V i +⃗a ∆ t 0=⃗v sin 30−9.8 ∆ t

∆ t=

1.81865 9.8

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∆ t=0.18 s e. Energies involved in the jump

E g1 is the gravitational energy at the start of the track.

E g 2 is the gravitational energy at the top of the ramp.

Ek 2 is the kinetic energy at the top of the ramp.

Assuming that friction and air resistance is negligible throughout the track:

E g1 =Eg 2 + E k2

Analysis A.

The calculated horizontal distance of the jump and the actual had a wide range and did not

match completely. The calculated distance of the jump should have been 1.45m while the actual jump resulted in being 0.14m, resulting in a difference of 1.31m. This huge inconsistency was caused due to out sources of errors not being accounted for in our calculations. The loss of total energy from the top of the track hugely impacted the energy the marble had at the top of the ramp resulting in a loss of speed.

This energy could have been lost to air resistance and friction as those two forces are always present throughout the track. Another source of the lost energy could be attributed to the collision of the fences on the track. Therefore, the energy of the marble was lost throughout the track and all the gravitational energy wasn’t completely transformed into kinetic energy.

B.

During this lab, the law of Conservation of Energy has been confirmed. The Law of Conservation

of Energy is that energy is always conserved in a closed/isolated system and is never created or destroyed (Nave). Whether energy was lost throughout the track, the total energy it had was still converted to not only kinetic energy but to thermal and energy and even noise/sound energy. The

12 mechanical energy produced through the conversion gravitational energy remained equal and it neither created energy nor destroyed it, but rather converted the energy to some other form like kinetic, thermal or sound. Therefore the law held true and has been confirmed as the starting gravitational energy was converted to all types of energy and helped bring the marble across the track.

C.

In this project, during observations and calculations, sources of error were present and caused

inconsistencies in our results. Some possible sources of error in this lab are collision into fences, air resistance and friction to the track and soaked tubes resulting in less traction with the ball. In this coaster project, fences around the track were constructed to keep the ball within the track. However, due to these fences, the ball collided with them and lost energy on its way down and lost huge amounts on the turns in this coaster. Also, an inevitable source of error being the energy lost to air resistance and friction on the track caused the total energy to be less than expected. Such sources of error could not be calculated and account for so it was considered to be negligible. Lastly, the tubing was soaked in boiling to help increase flexibility of the tubing but all tubes were not boiled equally. This resulted in the loss of traction in all tubes but more importantly, all pieces of tubing didn’t have the same traction at the end.

Therefore, sources of error were present in this lab and caused a difference to be in the actual results compared to the theoretical results.

D.

A clothoid loop is a distinct type of loop that is different than a natural vertically circular loop.

While a vertical circular loop has an equal radius, top to bottom, the clothoid loop instead has a larger radius at the bottom of the loop and a smaller one at the top. Due to the shape of a clothoid loop, the velocity of the cart can be way lower relative to a cart entering a circular loop but the clothoid loop can still be completed. Clothoid loops are used way more often in roller coasters due to a lower velocity being needed to complete the loop which reduces the force of gravity exerted on the passengers of the cart. Clothoid loops are used to safely ensure the loop is cleared at a low velocity, and to reduce the discomforting force of gravity that’s exerted on the passenger.

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References Nave, R. (n.d.). Retrieved December 23, 2017, from http://hyperphysics.phyastr.gsu.edu/hbase/conser.html (n.d.). Retrieved December 24, 2017, from

14 http://webs.bcp.org/sites/rjanke/rollercoaster/clothoid.htm