*Gravitational potential energy: gravitational potential energy is energy an object possesses because of its position in a gravitational field gained by height.
*Kinetic energy: kinetic energy is the energy of a body or a system with respect to the motion of the body or of the particles in the system.
Another force associated with roller coasters is friction. Fiction is the force which opposes the motion of the car, but also holds the car to the track. Although this friction is needed it is not wanted as it slows the car down. Efficiency calculates the percentage of useful energy output of the total energy input. So can tell us the amount of energy transferred to efficient forms and inefficient forms by friction.
Importance of efficiency and limitations
Efficiency is very important to ride manufacturers as it enables them to create tracks that reduce useful energy loss by friction. This reduces the energy needed at the start which helps them to reduce costs. Another important attribute that comes from finding efficiency is limitations. Efficiency enables ride manufactures to put limitations on their rides. This is important as insufficient energy input initially may make the car loose motion part the way through the ride. Excessive initial energy input may cause the ride to exceed other limitations such as track strength and G-force limits.
Importance of power and limitations
Power is very important to ride manufacturers as it is proportional to the energy consumption and therefore cost of the roller coaster. Because of this limitations are very important. A powerful motor increases the energy consumption and the costs but a weak motor takes a long time to transfer energy making rides potentially uninteresting, which also decreases profits. So a balance between these two limitations must be found.
Developments using efficiency
Using efficiency, developments into new track design and rollers can be made. These developments can be into even more frictionless tracks and rollers as these developments will be made to increase the useful energy transferred and therefore to decrease the running costs of the roller coaster. Another development using the efficiency of tracks is the track complexity. As the efficiency of the tracks increase the length of tracks could increase using the same input of power and therefore be more complex.
How to calculate power
Power is a measure of how quickly work can be done. It can be expressed Power = Energy / Time. To find out the power of anything we need two things, energy and time.
To work out energy of a roller coaster for the initial climb we can find the potential energy its gained.
Gravitational potential energy = Mass*Gravitational field strength
*Vertical height
GPE = M*G*H
So Power = M*G*H
Time
How to calculate efficiency
‘Energy conversion efficiency is the ratio between the useful output of an energy conversion machine and the input.’
We are able to find the efficiency of any part of the track although is complex, so to simplify the problem we shall calculated the efficiency of the entire track.
Efficiency = (Useful energy/Total energy)*100
Useful energy
E = (Ke/GPE)*100
Energy out is kinetic and energy in is GPE but GPE equals Ke so if we make energy in and out Ke then we can write the equation as:
E = (V/U)*100 U = Velocity in
V = Velocity out
To work out V:
V = Δs/Δt
Calculating power and efficiency
Ride name: G-Force
Data known: m = 1350Kg
t = 6.49s
g = 9.81
U = 20.25ms-¹
U Δs = 8m
V Δt = 1.74s
Data needed: h = ?
GPE = ?
P = ?
V = ?
Ef = ?
Finding h
mgh=0.5mv² ⇒ h=0.5g-¹v² ⇒
h = 0.5*9.81-¹*20.25² ⇒ h= 20.90m
Finding GPE
GPE=mgh ⇒ GPE = 1350*9.81*20.9 ⇒
GPE= 276789.15j
Finding P
P=E/t ⇒ P=GPE/t ⇒ P= 276789.15/6.49 ⇒
P= 42648.56w
Finding V
V=Δs/Δt ⇒ V=VΔs/VΔt ⇒ V= 8/1.74 ⇒
V= 4.60ms-¹
Finding Ef
Ef=(V/U)*100 ⇒ Ef= (4.60/20.25)*100 ⇒
Ef= 22.72%
Ride name: Apocalypse
Data known: m = 500Kg
g = 9.81
h = 54m
t = 15.12s
V Δs = 51m
V Δt = 2.72s
Data needed: GPE = ?
P = ?
U = ?
V = ?
Ef = ?
Finding GPE
GPE=mgh ⇒ GPE = 500*9.81*54 ⇒
GPE= 264870j
Finding P
P=E/t ⇒ P=GPE/t ⇒ P= 264870/15.12 ⇒
P= 17517.86w
Finding U
mgh=0.5mv² ⇒ v=√(2gh) ⇒
v=√(2*9.81*54) ⇒ v= 32.55ms-²
Finding V
V=Δs/Δt ⇒ V=VΔs/VΔt ⇒ V= 51/2.72 ⇒
V= 18.75ms-¹
Finding Ef
Ef=(V/U)*100 ⇒ Ef= (18.75/32.55)*100 ⇒
Ef= 57.60%
Bibliography
Physics of the roller coaster:
Fig.1
Fig.2
Fig.3 - sourced form:
Gravitational potential energy:
Quote on gravitational potential energy- sourced from:
Efficiency:
Quote on energy conversion efficiency - sourced from:
Kinetic energy:
Quote on kinetic energy - sourced from: