Anglican Church Grammar School
Physics
Amusement Park Physics:
Model Roller Coaster Design
Name: Brendan Ta
Subject: Physics
Class: 11 PH3
Teacher: LWB
Roller Coasters
1. Abstract / Hypothesis of a Good Roller Coaster
The roller coaster is a balance between safety and sensation. A good roller
coaster gives its riders thrill sensations that will make them want to ride it again
and again. The ride should still be as safe as possible, even though passenger
ride a coaster for the death defying thrill. The key to a successful coaster is to
give passengers the sensation of speed and acceleration, which all depends on
the speed control. A good roller coaster has hills, dips, loops, curves, straights,
and braking systems that obey the laws of physics.
The first experiment will show that mass does not affect the velocity. It will do this
by altering the mass of a toy car down a track from a fixed height. This will show
there is no relationship between the mass and velocity.
The second experiment will show the minimum velocity required to make a loop.
It will do this by altering the heights of a toy car along a track and measure the
minimum velocity at the top of the loop. This will be done for altering radii of
loops. It will prove there is a relationship between the minimum velocity and the
radius of the loop.
The third experiment will show the maximum velocity that can be reached on
hump before the ride becomes unsafe. It will do this by altering the heights of a
toy car along a track and measure the maximum velocity over the hump. This will
be done for altering radii of humps. This will prove that there is a relationship
between the maximum velocity and the radius of the hump.
By applying these theorem plus others to a roller coaster design it will be
possible to break down the critical points and determine if the design is exciting
and safe.
Brendan Ta 11.PH.3
2
Roller Coasters
2. Introduction / Physical Principles of Roller Coasters
Energy exists mainly as:
 Kinetic energy, energy due to motion
 Potential energy, the energy that is stored
A roller coaster works because of two things: gravity and the law of conservation
of energy, i.e. energy can change from one form to another but cannot be neither
created nor destroyed. A motor does the work to get you up the first hill and as
you travel up the hill, the coaster stores more and more potential energy. That
potential energy is converted into kinetic energy as gravity pulls it down the first
hill. The further you go down the hill, the greater the velocity, and the more
potential energy is changed into kinetic energy. The ride is fastest at the bottom
of the hill because more and more potential energy has been converted into
kinetic energy. In other words, the coaster trades height for velocity, or vice
versa. As you go up the next hill, kinetic energy is changed into potential energy
and the carriage slows down. The higher you go, the more energy is changed
and you feel the car slow down. The total energy neither increases nor
decreases; it just changes from one form to the other. However, some energy is
lost due to friction. This may include wind resistance, the rolling of wheels, and
heat or sound energy. The wheels reduce friction: it's easier to let something roll
than to let it slide. Roller coaster designers are aware of this, and therefore
make each successive hill lower than the previous one. Frictional losses can be
calculated by determining the difference between the total energy at any two
given points.
E lost = ME1 – ME2
E lost = (mgh1 + ½ mv21) - (mgh2 + ½ mv22)
For example, on the Cyclone, energy is lost travelling from the top of the first hill
to the top of the first loop.
E lost = ME1 – ME2
E lost = (mgh1 + ½ mv21) - (mgh2 + ½ mv22)
= (3500×9.80×32 + ½×3500×1.272) - (3500×9.80×21.7 + ½×3500×132)
= 1100423 – 1040060
= 6.0 × 104 Joules
The energy lost to friction was 6.0 × 104 J, travelling from the top of the first hill to
the top of the first loop.
On the Tower of Terror, there is a significant amount of energy lost due to friction
throughout the ride.
E lost = ½ mv2 - mgh
= ½×6.0×103×522 - 6.0×103×9.80×89
= 8112000 – 5059300
= 3.1 × 106 J
Brendan Ta 11.PH.3
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Roller Coasters
A unit that is used to describe forces we feel is the g. One g is equal to the force
of earth's gravity. Ironically, despite its name, gravity isn't a g-force, but g-forces
are measured in terms of what you feel when you are sitting still in the earth's
gravitational field. Engineers often use g's as a "force factor" unit. The force
factor gives people a way of comparing what forces feel like. On roller coasters,
the highest g-forces are felt just as the carriage enters a loop and also as it exits
a loop, as these are where the greatest changes in velocity and direction are.
The lowest g-forces are felt at the top of the loops, as this is where riders feel
“weightless”.
Just before entering a loop
Forces felt on loops
Brendan Ta 11.PH.3
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Roller Coasters
The physiological affect a roller coaster has on it riders must be calculated
because if there is unsafe g-forces fatal injuries could occur. G-forces can affect
the rider’s heart rate, stress blood vessels, injure necks and backs and cause
black out or even red outs. Larger riders when accelerating in any direction push
against the restrains of the coaster much more than a lighter rider. A riders
withstanding to these g-forces can depend on their size, age, pervious injuries
and medical disabilities, this may cause, bruising, breaking bones or even
internal injuries. A roller coaster designer must decide on how the restrains will
work and how they will pull or push against riders. If the restrains do not have
some cushion the riders maybe injured quite a lot more if they were hard.
Another consideration the roller coaster designer has to make is how tall a rider
can be to be safely harnessed in a ride. Many rides have height restrictions
because of this. Roller coaster harnesses often fit very tightly on the rider; this is
because if the rider is thrown around in there seat injuries result. Aged riders
must also be aware that the high g-forces because older people may have brittle
bones or heart conditions. A rider with a weak back or neck may also be injured
by the jerky turns that may occur on a roller coaster. Pregnant riders should also
avoid high g-forces and jerky movements. These are very dangerous to women
which are holding a baby because it does cause changes to a developing baby.
When g-forces occur in high amounts or for long amounts of time the body is
affected in many ways.
Collins Concise Encyclopaedia of Astronautics (1968), pp 3-4, states:
“A man in fact can withstand very high accelerations provided their
duration is cut as their magnitude increases.”
Humans can instantaneously sustain up to 500g in car accidents or falls on hard
surfaces and 12g when diving from a springboard into water. However, for safety
and legal reasons, roller coasters should not exceed 3g’s for more than a few
seconds when entering a loop. There are several types of g-forces that are felt
by humans.
Life Support: G-Forces states:
(http://nasaui.ited.uidaho.edu/nasaspark/safety/lifesupport/gforces.html)
“The name of the G-Force is based on the direction of force with
respect to the long axis of the body. The inertial force is the primary
factor that produces physiological effects on the body during
acceleration.”
Brendan Ta 11.PH.3
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Roller Coasters
Positive g-forces occur when an accelerative force is applied from head to foot
on the body, making the rider feel pushed down in the seat. As the blood pools
away from the head, a condition called grey-out may occur.
Negative g-forces occur when an accelerative force is applied from foot to head
on the body, lifting the rider up out of the seat. The blood pools in the head, and
with the increase pressure, vision turns red (redout), and the eyeballs feel like
they are popping out.
Transverse g-forces occur when the accelerative forces act across the body in a
back to chest or chest to back direction.
Lateral g-forces occur when accelerative forces act across the body in a side-toside direction. Several factors affect the strength of lateral G forces: the speed of
the train, the tightness of the curve, and the amount of banking.
As the g-forces climb toward 5g, you feel five times your weight and sink further
into your seat. You can no longer see colour, only black and white. Next your
field of vision begins to shrink and it feels like you are looking through a pipe.
Your peripheral vision disappears as the visual pipe's diameter gets smaller and
smaller. You feel heavier and heavier as the g-force increases. In a flash you see
black. You have just experienced “blackout”. You remain unconscious until the
g-force declines and the blood returns to your brain. Extended periods of
blackout will lead to fainting, permanent brain damage, chronic seizures, heart
failure, and death. This is avoided as much as possible by designers and
insurance companies, and would limit repeat potential riders. Most roller coasters
keep the g's felt under 5 g's on an inside loop or the bottom of a dip after a hill.
When a rider travels over a hill very quickly, he experiences negative g's. A
negative g is the multiple of a person's weight that is needed to keep a rider in his
seat. Negative g's also force the coaster car to try to come up off the track.
Tony Wayne states:
(http://141.104.22.210/Anthology/Pav/Science/Physics/book/)
“Negative g's are a rider's heaven and a designer's nightmare.
Negative g's are avoided as much as possible.”
A negative g however, has a different effect on a rider compared to a positive g.
Both negative and positive g's can cause a rider to pass out. However, negative
g’s cause a “redout”. A redout occurs when there is too much pressure on the
brain caused by a surplus blood flowing into the head. The high pressure can
cause blood vessels to burst and kill the rider. This is a sure way to limit the
number of repeat riders.
Brendan Ta 11.PH.3
6
Roller Coasters
Weightlessness and heaviness are linked with the normal force; but have little to
do with gravity. The normal force is smaller at the top of the loop and larger at the
bottom of the loop. The normal force is large at the bottom of the loop because
the normal force must be greater than the outward gravity force in order for the
net force to be directed inward. At the top of the loop, the gravity force is directed
inward and thus, there is no need for a large normal force in order to sustain the
circular motion. The fact that a rider experiences a large force exerted by the
seat upon her body when at the bottom of the loop is the explanation of why she
feels heavy. In reality, she is not heavier; only experiencing the large magnitude
of force which is normally exerted by seats upon heavy people while at rest. Gforces can be calculated by dividing the acceleration by 9.8ms -2, or by taking
readings from the vertical and horizontal accelerometers, which in turn can be
converted into accelerations by multiplying it by 9.8ms-2. However, when
determining the g-force vertically, 1g must be subtracted from the final result to
account for gravity, which is always present.
Tower of Terror
Horizontal accelerometer reading going up was 0.7g.
aH = 0.7g
Therefore, aH = 7ms-2
Just before the car enters the upward curve, the vertical accelerometer was 3.5g.
acA = 3.5g
Subtract 1g
acA = 2.5g
Therefore, aH = 24.5ms-2
T
o
w
e
r
o
f
acA = 2.5g
aH = 0.7g
T
e
r
r
o
r
Brendan Ta 11.PH.3
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Roller Coasters
Cyclone
At top of loop, g-force = ((v2/r)-g) ÷ g
= ((12.52/8)-9.8) ÷ 9.8
= 1.0
At bottom of loop, g-force = ((v2/r)+g) ÷ g
= ((20.62/7.9)+9.8) ÷ 9.8
= 6.5
Gravitron
R Top = 5.6m
1m
R = 5m
1m
R Bottom = 4.4m
30°
Χ = tan 30°
Χ = 0.6m
Therefore, R Top = 5.6m, R Bottom = 4.4m
χ
1m
30°
acA Top = 42r/T2
= (42 × 5.6) ÷ 2.6052
= 32.6ms-2
= 3.3g
acA Bottom = 42r/T2
= (42 × 4.4) ÷ 2.6052
= 25.6ms-2
= 2.6g
Brendan Ta 11.PH.3
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Roller Coasters
3. Diagram of Roller Coaster Design
Total length of rail:
Mass of train:
Passenger Capacity:
Max speed:
Lift-up mechanism:
Ride time:
Conveyor system:
approx. 2200m
3000kg
20 persons
28ms-1 (100kmh-1)
Conveyor chain
approx. 4 minutes (includes loading and unloading)
Lift-up speed: 2 ms-1
Angle of lift: 30°
“The Spine Melter”
Brendan Ta 11.PH.3
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Roller Coasters
4. Analysis of Roller Coaster Design
The “Spine Melter” has been designed to give riders a death-defying thrill
sensation while at the same time being safe. It incorporates a steep first hill
climb and fall, slopes, a helix, two vertical loops, and a banked curve.
The train, with a mass of 3000kg, exits the station and is pulled up the first hill
with height of 40m by a conveyor chain at a steady 2 ms -1. When it reaches the
peak, the train’s centre of mass will have a velocity of zero.
ME = mgh + ½ mv2
ME = (3000×9.8×40) + 0
ME = 1,176,000J
v=0
v = 2ms-1
v = 28ms-1
40m
As it travels down the incline, it reaches its maximum speed of 28ms -1 and enters
the first slope (15m) with a velocity of 22.14ms-1 and the second slope (10m) at
24.25ms-1.
1st slope
ME = mgh + ½ mv2
1,176,000 = (3000×9.8×15) + (0.5×3000× v2)
1500 v2 = 735000
v2 = 490
v = 22.14ms-1
2nd slope
ME = mgh + ½ mv2
1,176,000 = (3000×9.8×10) + (0.5×3000× v2)
1500 v2 = 882000
v2 = 588
v = 24.25ms-1
Brendan Ta 11.PH.3
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Roller Coasters
At B, about 4% of energy to friction, leaving 1128960J left. The train then enters
a helix with a radius of approx. 5m at about 20.48ms-1.
ME = mgh + ½ mv2
1,128,960 = (3000×9.8×17) + (0.5×3000× v2)
1500 v2 = 629160
v2 = 419.44
v = 20.48ms-1
Using circular motion
ac = v2/r
ac = 20.482/5
ac = 84ms-2
It continues on a steady descend for about 200m at the same velocity. At C,
about 10% of energy to friction, 1058400J left. It enters the first loop with at
about 26.2ms-1 and at the top has a velocity of 17.7ms-1. The loop has a radius
of 17m at the bottom, 10m at the top, and a height of 28m.
ME = mgh + ½ mv2
1,058,400 = (3000×9.8×1) + (0.5×3000× v2)
1500 v2 = 1029000
v2 = 686
v = 26.2ms-1
G-forces
At top of loop, g-force = ((v2/r)-g) ÷ g
= ((17.72/10)-9.8) ÷ 9.8
= 2.2
At bottom of loop, g-force = ((v2/r)+g) ÷ g
= ((26.22/17)+9.8) ÷ 9.8
= 5.12
At D, approx. 15% of energy has been lost to friction, 999600J left. The train
enters the loop with a velocity of about 25.8ms-1 and at the top the velocity will be
about 16ms-1. The loop has a radius of 14m at the bottom, 9m at the top, and a
height of 26m.
ME = mgh + ½ mv2
999,600 = 0 + (0.5×3000× v2)
1500 v2 = 999,600
v2 = 666.4
v = 25.8ms-1
Brendan Ta 11.PH.3
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Roller Coasters
G-forces
At top of loop, g-force = ((v2/r)-g) ÷ g
= ((162/9)-9.8) ÷ 9.8
= 1.9
At bottom of loop, g-force = ((v2/r)+g) ÷ g
= ((25.82/14)+9.8) ÷ 9.8
= 5.9
After exiting the second loop, about 20% of the total energy has been lost due to
friction, with 940800J left. It then travels over a 15 m slope at 18.25ms-1.
ME = mgh + ½ mv2
940,800 = (3000×9.8×15) + (0.5×3000× v2)
1500 v2 = 499800
v2 = 333.2
v = 18.25ms-1
The train, with its velocity already greatly reduced due to friction, enters a banked
curve tilted at 30° to the horizontal.
R = (18.152×tan30)/9.8
R = 19.6m
G-Forces
g = 1/sin30
g=2
Brendan Ta 11.PH.3
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Roller Coasters
Finally, the brakes are applied to the train to bring it to a gradual halt. The total
energy is 940800J, and the train must still travel 100m until it reaches the station.
Braking force
W = ∆E
W = Fs
940,800J = 100F
Therefore, F = 9408N
The average braking force is 9408N
Overall, 25% of the total mechanical energy was lost due to friction.
5. Conclusion
The Spine Melter compared quite well with the Cyclone and Cyclone, which
reached speeds of 23.6ms-1 and 24.2ms-1respectively. The Superman Krypton
Coaster at Six Flags Fiesta, Texas, is a similar roller coaster to the Spine Melter.
It can reach higher velocities (31.3ms-1), greater heights (51.2m), but its duration
much shorter in length (2:35). It is arranged in 3 trains with 8 cars per train.
Riders are arranged 4 across in a single row for a total of 32 riders per train. It
also has a very high and steep first hill, and many inversions. It seems that the
higher the first hill is and how far it drops, the greater the velocity reached by the
train. However, most would feel the greatest thrills occur during the loops and in
the dives. Riders feel sudden changes in velocity and direction, which have a
lasting effect on them. It is important that roller coasters do not exceed 5g’s or
low negative g-forces for any more than an instant. Coaster designers try to
avoid these as much as possible, as extended redouts and blackouts can result
in deaths, which will severely affect the number of repeat and potential riders,
and also cause many legal battles. Above all, roller coasters should be a
balance between safety and sensation.
Superman Krypton Coaster at Six Flags Fiesta, Texas
Brendan Ta 11.PH.3
Cyclone, Dreamworld
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Roller Coasters
Bibliography
Collins Concise Encyclopaedia of Astronautics (1968), pp 3-4
http://www.esc20.net/etprojects/formats/webquests/spring2001/holmeshealth/Am
usementParkPhysics/default.html
http://141.104.22.210/Anthology/Pav/Science/Physics/book/
http://www.learner.org/exhibits/parkphysics/
http://nasaui.ited.uidaho.edu/nasaspark/safety/lifesupport/gforces.html
http://www.thinkquest.org/library/lib/site_sum_outside.html?tname=C005075F&ur
l=C005075F/English_Version/coasters.htm
http://www.chebucto.ns.ca/~ak621/CEC/Co-Phys.html
http://www.rcdb.com/installationdetail605.htm
http://www.ed.uri.edu/smartprojects/rcoaster.html
Brendan Ta 11.PH.3
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Roller Coasters
Experiment 1
Aim: To show that the mass does not affect the velocity of a roller coaster when
released from a constant height.
Introduction:
Physics laws show that mass does not affect velocity. This means if a roller
coaster carries a heavier load this will not affect how it runs the course aiming it
safe no matter what the load. Mass does however corse more friction and this
would mean small variation in results. This means that there is no relationship
between mass and velocity.
It is a common misconception of riders that if they have a heavier load their ride
will change in speed. This experiment will disprove this misconception. The track
and toy-car will be set up and released from a fixed height along with Data Studio
using and picket fence and photogate to measure the velocity. Using this set up
will show that there is no change except the slight changes from friction on the
velocity when the mass is changed thus proving there is no relationship between
mass and velocity.
Procedure:
1. Height was chosen
2. Track and Data Studio using picket fence and photogate was configured
3. Toy car was released from fixed height along track.
4. The velocity was measured at the bottom of the track
5. This was repeated for each varying mass three times.
6. Results were put into an appropriate table.
7. Average was recorded.
Car
Incline
49cm
Photogate
Brendan Ta 11.PH.3
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Roller Coasters
Results:
Mass of toy car= 37g height of release=49cm
1
2
3
4
Mass of
toy-car
and
weights
(g)
47
57
87
97
Velocity
(ms-1)
Velocity
(ms-1)
Velocity
(ms-1)
Average
Velocity
(ms-1)
1.39
1.39
1.41
1.41
1.39
1.39
1.39
1.39
1.39
1.39
1.39
1.41
1.39
1.39
1.40
1.40
The table shows the toy car’s velocity at the bottom of the ramp. The readings
are very similar meaning they are both precise and accurate. The velocity
changed very little proving there is no relationship between velocity and mass.
Conclusion:
This experiment proved that there is no relationship between mass and velocity
at a fixed height.
Discussion:
This experiment demonstrated that there is no relationship between mass and
velocity at a fixed height. The original theory that mass does not affect velocity is
true.
Many errors may have also affected the readings:
 Releasing of the car non-uniformly.
 Car moving out of rails.
 Friction.
Improvements could be made to increase the accuracy of the experiment. Use of
a better way of releasing the cars uniformly, perhaps a gate that can be lifted so
the car is not pushed faster or dislodged from rails or starting point.
Brendan Ta 11.PH.3
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Roller Coasters
Experiment 2
Aim: To prove the relationship between the minimum velocity and the radius of a
loop.
Introduction:
Velocity is exponentially proportional to the radius of a loop. The velocity gained
at the top of the loop must be high enough to maintain enough g-forces so that it
can stay on the track, this is called the minimum velocity. The theory states V 2min
= (Rg) thus meaning the minimum velocity needed is exponentially related to the
radius of the loop. If the cart does not maintain this minimum velocity at any part
along the loop it will fall off or stop.
Roller coasters rely on big turns and high g-forces to excite riders. It is common
for riders to think that not matter how fast they are going they will always make a
loop. Physics theory however states that a minimum velocity must be reached for
the ride to make the entire loop. When a roller coaster is being designed,
designers must used physics to find if the ride has enough speed and energy to
make a loop. If this speed is too slow the coaster may stop or even fall off.
The experiment will display the relationship between the minimum velocity and
the radius by realising the toy car at varying heights to find the minimum velocity
required to make the loop. This will be done with differing radii. This will then
prove the exponential relationship between minimum velocity and radius of a
loop. By knowing this it can then be applied to our roller coaster design.
This experiment will prove the relationship between the velocities required to
complete loops of differing radii. A car and track formed into a loop will be used
along with Data Studio to find the velocity.
Procedure.
1. Track and photogate and picket fence were calibrated.
2. Data Studio configured.
3. Four differing radii were chosen.
4. Toy car was released at differing heights on the track.
5. Minimum velocity was recoded at the top of the loop.
6. Repeated 3 times for each loop of differing radii.
7. Results put into table.
8. Average of results calculated and recorded
Brendan Ta 11.PH.3
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Roller Coasters
Car
Start ramp
Photogate
Loop
Results:
1
2
3
4
Radius
(m)
Velocity 1
(ms-1)
Velocity 2
(ms-1)
Velocity 3
(ms-1)
0.070
0.105
0.160
0.190
0.870
1.06
1.28
1.32
0.810
1.04
1.30
1.39
0.880
1.07
1.30
1.42
Average
Velocity
(ms-1)
0.850
1.06
1.29
1.37
The table shows the velocity of the car at the top of the loop of differing radii. The
table shows the velocities were very similar meaning that they were both
accurate and precise. As the radius increased so did the velocity adhering to our
theory of exponential relationship.
Sample Calculations.
V2= (Rg)
V= √ (Rg)
V= √ (0.105 x 9.80)
V= 1.01 ms-1
(Experiment result was 1,06 thus meaning a 5% of error)
Conclusion:
This experiment proved the exponential relationship between minimum velocity
that the car needed to go around the radius of the loop.
Brendan Ta 11.PH.3
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Roller Coasters
Discussion:
This experiment demonstrated that when the radius of the loop is made bigger
the minimum velocity needed to complete the loop increase. The theory that the
minimum velocity is exponential to the radius is a correct theory.
Many errors may have also affected the readings:
 Releasing of the car non-uniformly.
 Car moving out of rails.
Improvements could be made to increase the accuracy of the experiment. Use of
a better way of releasing the cars uniformly, perhaps a gate that can be lifted so
the car is not pushed faster or dislodged from rails or starting point.
Brendan Ta 11.PH.3
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Roller Coasters
Experiment 1
Aim: To find the relationship between the maximum velocity and radius of a
hump.
Introduction:
The maximum velocity is exponentially related to the radius of a hump. The
velocity at the top of the hump must be lower than the maximum velocity. Physics
theory statesV2= (Rg) where the maximum velocity is exponentially proportional
to the radius of the hump. If this velocity is to high the object may be come
derailed or even air-born meaning the ride would be very dangerous.
Most riders assume that no matter what velocity they are travelling at they will
always make the hump safely. As according to physics theory there is a limit to
this velocity and this experiment will prove this theory. Designers of roller
coasters must find this maximum velocity and must not come close to it or
exceed it or the ride becomes very dangerous.
This experiment will show this exponential relationship between the maximum
velocity and the radius of the loop. It will do this using a toy car and track set up
with a hump along with Data Studio to measure the velocity at the top of the
hump.
This experiment will use the toy car and hump at varying heights to find the
maximum velocity over the hump and also varying radii. The maximum velocity
will be measured using a picket fence and photogate set up. Using this
procedure the exponential relationship between the maximum velocity and radius
of the hump can be proved.
Procedure:
1. Track and photo gate and picket fence calibrated.
2. Data Studio configured
3. Four differing radii were chosen for the humps.
4. Toy car released from carrying heights on the track.
5. Maximum velocity was measured at the top of the hump
6. Repeated three times for each different hump.
7. Results put into table
8. Average taken and results graphed in excel.
Brendan Ta 11.PH.3
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Roller Coasters
Car
Results:
Starting
ramp
Photogate
Hump
1
2
3
4
Radius
(m)
Velocity
(ms-1)
Velocity
(ms-1)
Velocity
(ms-1)
0.045
0.060
0.090
0.115
0.65
0.73
0.97
1.09
0.61
0.75
0.99
1.07
0.63
0.75
0.99
1.12
Average
Velocity
(ms-1)
0.63
0.74
0.98
1.09
The table shows that the readings were very similar meaning they were both
precise and accurate. It shows that as the radius of the hump raised so did the
maximum velocity signifying at exponential proportionality.
Sample Calculations:
V2=(Rg)
V= √ (Rg)
V= √ (0.045 x 9.80)
V= 0.66 ms-1
(Experiment result was 0.63 giving a 5% error)
Conclusion:
The experiment proved the theory that maximum velocity is exponentially
proportional to the radius of a hump. Therefore it can be said that velocity is
close to directly proportional to the radius of the hump.
Brendan Ta 11.PH.3
21
Roller Coasters
Discussion:
This experiment demonstrated that when the radius of the hump is made bigger
the maximum velocity that is required to stay on the track increased. The theory
that the maximum velocity is exponential to the radius is a correct theory.
Many errors may have also affected the readings:
 Releasing of the car non-uniformly.
 Car moving out of rails.
Improvements could be made to increase the accuracy of the experiment. Use of
a better way of releasing the cars uniformly, perhaps a gate that can be lifted so
the car is not pushed faster or dislodged from rails or starting point.
Brendan Ta 11.PH.3
22