Roller Coaster Physics

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Roller Coaster Physics!

• All the math you need to know so that you dont
  harm your passengers!

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What we are assuming

• No Friction
• To ease the burden of your calculations, we will
  assume that friction is negligible (not worth
  counting).
• No air resistance
• We will not count in the effects of air
  resistance, either.
• Energy will not be wasted as heat energy during
  energy transformations.

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Calculating Velocity

• You will need to calculate the velocity of your
  train of 1 to 4 cars at
• Beginning and end of each hill
• Beginning, top, and end of each loop
• In the middle of each lateral turn

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Calculating Velocity

• Energy is Conserved!
• PE before KE before PE after KE after
• Start your calculations from the top of the first
  hill, assume an initial speed of 1 m/s

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Example

• point 1
• Height 50 m
• Velocity 1 m/s
• Mass 800 kg

• point 2
• Height 15 m
• Velocity ? m/s
• Mass 800 kg

PE 1 KE 1 PE 2 KE 2
mgh 1 ½ mv2 1 mgh 2 ½ mv2 2
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Example
mgh 1 ½ mv2 1 mgh 2 ½ mv2 2
(800kg)(10m/s2)(50m) 1 ½ (800kg)(1m/s)2 1
(800kg)(10m/s2)(15m) 2 ½ (800kg)v2 2
400,000J 400J 120,000J (400kg)v2
400,400J 120,000J (400kg)v2
280,400J (400kg)v2
701 m2/s2 v2
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26.476 m/s v at point 2
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Lets make it easy!

• If you know that ME PE KE, and that energy is
  conserved, you should know that ME should always
  be the same value!
• What was the ME from the last example?
• ME 400,400J
• So, for your project
• Just figure out the ME at the top of the first
  hill use that number to help you find velocity!

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Calculating Work (J)

• Work ME
• ME PE KE, so Work PE KE
• The work it takes to start or stop an object from
  moving is equal to the ME!
• For starting your roller coaster, work PE KE
  at the top of the 1st hill
• To stop work PE KE before you slow it down.

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Power Electricity

• Power Work / time
• Power is measured in watts.
• 1 Kilowatt (KW) 1000 watts

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Electricity use for 1 ride

• P W/t
• W the total energy to start and stop the ride
  (in Joules J)
• t the time of 1 ride (in seconds!)
• Convert this into Kilowatts you have the
  electricity for 1 ride!

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Time

• Use t d/s
• d distance between 2 points
• s the average speed between the 2 points
• Example

Average speed (1m/s 37 m/s)/2
(38m/s)/2 19 m/s
t d/s t 50m /19 m/s t 2.63 s
Point 1 v 1m/s Point 2 v 37 m/s
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d 50 m
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Time for 1 ride

• Time for 1 ride sum of all segment times and
  the loading / unloading time.
• How many rides/ hour?
• 60 minutes/ time of 1 ride
• How many rides/ day?
• How long is a day at your theme park?
• Divide this time by the time of 1 ride, make sure
  and convert from seconds to minutes and so on

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Cost per ride

• Find the power (in KW) and the time (in hours).
• The cost of electricity is about 0.22 per
  Kilowatt hour.
• Cost per ride Power X time X 0.22
• Cost per day will depend on how many runs you
  have.

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Other considerations

• How many people are in each train of your roller
  coaster (this depends on mass!)?
• How many trains do you have? You can have more
  than 1 running at the same time.