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Roller Coaster Energy

and

A Little Math

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Here is that same video with Mr. K (use the script below)

If you have been on a rollercoaster you know the anticipation of slowly being lifted up to the top of that first hill. Right? Your stomach is in a little knot, your adrenalin is causing your heart to beat faster. You’re starting to wonder if you will get out of this alive.

Then you finally make it to the top...

…The roller coaster moves slowly over the edge but then it usually stops for a few seconds. If your in one of the front cars you are left hanging over the edge not knowing when the train was going to suddenly drop…the suspense is killing you!

Engineers design roller coasters to purposely give you all those feelings of thrill and terror. But because rollercoasters do not have a motor to give them the energy to scare you, they rely on different types of energy to complete your thrill ride.

Let’s take a look

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Potential Energy

Resting

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Kinetic Energy

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Kinetic and Potential Energy

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Kinetic and Potential Energy

Gravity!

Gravity is the source of

Kinetic and Potential Energies

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Accelertion

due to

Gravity

Gravity = 9.8 m/s

2

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Gravity!

Gravity = 9.8 m/s

2

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A Little Math

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Potential Energy (PE):

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Potential Energy (PE):

PE = mgh

PE = (mass) x (acceleration due to gravity) x (height)

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120 m

75kg

PE = m g h

Click HERE

To see the math

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120 m

PE = m g h

PE = 75 x 9.8 x 120

75kg

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120 m

PE = mgh

PE = 75 x 9.8 x 120

75kg

88,200

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120 m

PE = mgh

PE = 75 x 9.8 x 120

75kg

joules

PE = 88,200

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Kinetic Energy:

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Kinetic Energy:

KE = (1/2 mass) x (velocity) x (velocity)

KE = 1/2 mv²

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120 m

KE = 1/2 mv²

Mass (m) =75kg

Velocity (v) = 40 m/s

B

A

Click HERE

To see the math

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120 m

B

A

KE = 1/2mv²

Mass (m) =75kg

Velocity (v) = 40 m/s

KE = 1/2 (75) x 40 x 40

KE = 60,000 J

KE = 37.5 x 1600

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Why is knowing the energy

so important?

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Ouchy!

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Velocity ?

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V = rg

√

Velocity (v) = the square root of the (radius) x (the acceleration due to gravity)

Velocity ?

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V = rg

√

9 meter

radius

Radius (r) = 9 m

Gravity (g) = 9.8 m/s

2

Velocity ?

Click HERE

To see the math

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9 meter

radius

V = rg

√

Radius (r) = 9 m

Gravity (g) = 9.8 m/s

2

Velocity ?

√

V = 9 x 9.8

V = 88.2

√

88.2

Click Here

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9 meter

radius

V = rg

√

Radius (r) = 9 m

Gravity (g) = 9.8 m/s

2

V = 9 x 9.8

√

V = 88.2

√

V = 9.39 m/s

Velocity ?

9.39

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9 meter

radius

V = rg

√

Radius (r) = 9 m

Gravity (g) = 9.8 m/s

2

V = 9 x 9.8

√

V = 88.2

√

V = 9.39 m/s

Velocity ?

?

a

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?

Height of point a =

a

h = h + 1/2

V

g

2

a

b

divided by the acceleration due to gravity.

height of point b + 1/2 the velocity squared

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?

Height of point b is twice the radius = 18 m

a

b

Velocity (v) = 9.39 m/s

9 meter

radius

Click HERE

To see the math

Gravity (g) = 9.8 m/s

2

h = h + 1/2

V

g

2

a

b

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a

b

9 meter

radius

h = 18 + 1/2

(

9.39

9.8

2

)

h = 18 + 1/2

(

88.17

9.8

)

h = 18 + 1/2

(

9

)

h = 18 + 4.5

h = 22.5 meters

22.5 m

h = h + 1/2

V

g

2

a

b

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22.5 m

Click HERE

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22.5 m

We made it!

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G-force

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G-Force

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G-force (negative/positive)

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Roller Coaster Skater

Don’t try this..it’s very dangerous!

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This is the end of this lesson.

Continue to the next lesson.

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The following slides contain optional content ideas

(Just some “free thinking” from Mr. K)

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Optional topics of discussion

Calculus �To accurately model every component of roller coaster design, a branch of math called calculus is needed. Calculus is used to create and analyze curves, loops, and twists along the roller coaster track. It helps with slope calculations and finds the maximum and minimum points along the track.

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The total energy in the system (in the roller coaster) is found by finding the PE of the starting point (the highest point where the marble begins).
PE = mgs (mass X acceleration of gravity X height)
If height at point A is 4 ft and the marble weighs 5 grams and 32 ft/second squared

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Video (Car drives through a loop)

https://www.youtube.com/watch?v=wiZoVAZGgsw

Hotwheel life size cars drive loop

https://www.youtube.com/watch?v=gdfvwBuYJ50

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consider Isaac Newton’s First Law, “that an object in motion tends to stay in motion with the same speed and in the same direction unless acted upon by another force.” When you are pressed against a car door as the car goes around a curve, you may feel pushed outward, much as you would on a merry-go-round. Really, there is no force pushing you outward. Your body wants to go in a straight line, but the car holds you in, acting as the other force to make you go in a curved line. On the merry-go-round, you are holding yourself onto the ride, exerting a force that keeps you from flying off to travel in a straight line.

, “What happens when you are riding in a car and the driver brakes really hard?”

Put a ball or a penny in a bucket and swing it in a small circle, so that the cup goes upside down like a loop-the-loop roller coaster. Swing it fast enough to keep the item from falling out.

Ask kids: “Why didn’t the ball fall out? What would happen if I swung the mini-bucket more slowly?” (Gravity would pull the item toward the ground). Ask kids, “Now that we tried it with a ball or penny, what do you think will happen with a ping pong ball or water?”

This is centrifugal

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Calculating the G-forces on the drivers is actually quite simple. We just need to know the radius of the turns and the speed of the cars

If a race car is going 230 mph around a curve with a radius of 750 ft, calculate g force like this:

Convert mph to f/s

-230 mph is 337 feet per second (f/s).

Now find the acceleration of the car (The amount of acceleration is equal to the velocity of the car squared divided by the radius of the turn)

-(337 f/s)2 / 750 feet = approximately 151 f/s2.

-Acceleration of the car is 151 f/s2.

The acceleration due to gravity (1 G) is 32 f/s2. or 9.8 m2
Now do the math

-151 / 32 = 4.74 Gs experienced by the drivers.

http://science.howstuffworks.com/science-vs-myth/everyday-myths/question633.htm

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Lets calculate the gforce of a rollercoaster

G force =

Acceleration in ft/s²

32

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What if we don’t know the radius but we know the g force and the speed of the coaster?

What we know

Rollercoaster Speed:50 mph

G-Force:4.0

Substitute the variables and solve for r (radius)

4 (g fore) =

G force =

Acceleration in ft/s²

32

(72.89 ft/s)² / r(radius)

32

Radius =?

Imagine: You are an engineer. A customer needs you to Design a roller coaster that has a speed of 50 mph and they want two of the curves to reach 5 g’s. You will need to figure out what the radius of these curves will be.

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Loops (math)

To find the lowest speed that the coaster needs to make it around the loop, we use the equation:

The “V” in this equation represents the smallest speed that the car needs, ”critical velocity”, while “r” is the radius of the loop. The letter “g” is the acceleration due to gravity, which is 9.8 m/s².

V = rg

√