1. UNIVERSITY OF SAN DIEGO
UPWARD BOUND 2016
THE PHYSICS OF
ROLLER COASTERS
UNIVERSITY OF SAN DIEGO
TOM SKELTON
CONNER HOUGHTBY

2. WHAT DO YOU HOPE TO LEARN?

3. MY TWO QUESTIONS:
1. WHAT PULLS THE TRAIN?
2. WHY NOT CIRCLES FOR THE LOOPS?

4. COURSE GOAL: BASICS OF ROLLER COASTERS
COURSE LEVEL:
MOST OF PHYSICS
SOME OF THE MATH
OUTLINE:
DEFINITIONS
NEWTON’S LAWS
APPLICATIONS (!)

5. DEFINITIONS WE WILL NEED
VECTOR: HAS MAGNITUDE AND DIRECTION
SPEED: HOW FAST IT’S MOVING
VELOCITY: SPEED AND DIRECTION
ACCELERATION: HOW FAST VELOCITY
IS CHANGING - A VECTOR
THIS MEANS THAT A CAR IS ACCELERATING IF IT IS
SPEEDING UP, SLOWING DOWN, OR ROUNDING A CURVE
YOU NEED TO REMEMBER THESE

6. NEWTON’S LAWS
An object at rest tends to remain at rest, and an object in motion
tends to remain in motion in a line, unless acted on by a NET force.
2. Net Force equals mass x acceleration
3. If Object A exerts a force on Object B, then Object B exerts
a force on Object A which is equal and opposite.    
The second law is written F NET = m a or a = F NET / m

7. The NET FORCE is the VECTOR SUM of all forces

8. Examples of forces
Fg = Gravitational force – weight, pointed downwards
FN = Normal Force – support force from tabletop, rails, etc.
“normal” means the direction is perpendicular.
Ff = Friction force – direction is along surface, rails, etc.
FD = Drag force – opposite direction as velocity
FT = Tension force – pull by a string, rope, etc.

9. forces act on the box (car)?
A FREE BODY DIAGRAM HELPS ANALYSIS
What force or
forces act on the box (car)?

10. A FREE BODY DIAGRAM HELPS ANALYSIS    
F g is fixed; and F NET = F g + FN must point in the direction as a

11. MOTION IN A CIRCLE AT CONSTANT SPEED:
Is it accelerating?

12. MOTION IN A CIRCLE AT CONSTANT SPEED:
Is it accelerating?
YES!! THE
VELOCITY
IS CHANGING!

13. MOTION IN A CIRCLE AT CONSTANT SPEED: Is it accelerating?
THE ACCELERATION IS TOWARDS THE CENTER OF THE CIRCLE

14. MOTION IN A CIRCLE AT CONSTANT SPEED:
Is it accelerating? YES; | a | = v2/R  
| a | = v2/R
v = speed
R = radius
THE ACCELERATION IS TOWARDS THE CENTER OF THE CIRCLE

15. APPLY WHAT WE HAVE LEARNED TO DRIVING IN CIRCLE
ON A BANKED ROAD d
FREE-BODY DIAGRAM
SKETCH
SKETCH
HORIZONTAL VIEW.
CAR COMING AT YOU.
NO FRICTION. FN
TOP VIEW.
TRACK IS
BANKED. a
F NET Fg
IN NORMAL DRIVING, FRICTION WOULD MEAN THAT SPEED v DOESN’T HAVE TO BE EXACT.

16. APPLY WHAT WE HAVE SO FAR TO TOP OF LOOP:
FREE-BODY DIAGRAM d
F NET = m a
F NET = m v2/ R
F NET = Fg + FN
Fg + FN = m v2/ R
FN = m v2/ R - Fg
SKETCH FN Fg a
F NET
What you feel is FN ,
the force on your body
FN MIGHT BE NEAR ZERO; IF NEGATIVE …

17. APPLY WHAT WE HAVE SO FAR TO BOTTOM OF LOOP:
FREE-BODY DIAGRAM d
F NET = m a
F NET = m v2/ R
F NET = FN - Fg
FN - Fg = m v2/ R
FN = m v2/ R + Fg
SKETCH FN a
F NET Fg
FN WILL BE MUCH GREATER THAN Fg; FURTHERMORE… …

18. ENERGY IN A ROLLER COASTER
GENERAL REMARKS ON ENERGY:
ALWAYS CONSERVED – ONLY CONVERTED FROM
ONE FORM TO ANOTHER
SEVERAL FORMS:
KINETIC – ASSOCIATED WITH MOTION
GRAVITATIONAL POTENTIAL – HEIGHT
THERMAL – BASICALLY SAME AS HEAT
ELECTRIC
CHEMICAL – SUCH AS A BATTERY
OTHERS

19. ENERGY IN A ROLLER COASTER
THE TOTAL ENERGY STAYS CONSTANT. POTENTIAL ENERGY CHANGES FORM TO KINETIC ENERGY K P K K K P P P

20. ENERGY IN A ROLLER COASTER
K = ½ mv2
P = mgh
ETOT = K + P = constant
Since there is really some friction, some energy is converted to heat. Ignore… K P K K K P P P

21. g is the acceleration due to gravity; same for any object.
K = ½ m v2
m = mass; often confused with weight, since they are similar in everyday life. Mass is the total amount of material. Weight is the gravitational pull on it, Fg.
Fg = m g
g is the acceleration due to gravity; same for any object.
v is the speed, as before.
P = m g h
h is height above baseline.
ETOT = K + P = constant (ignoring loss to heat)

22. ENERGY IN A ROLLER COASTER
K = ½ mv2
P = mgh
ETOT = K + P = constant
Since there is really some friction, some energy is converted to heat. Ignore… K P K K K P P P

23. A CAR IS TO BE RELEASED AT POINT “A.”
WHICH TRACK DESIGN WILL HAVE IT GOING THE FASTEST AT “B?” A B

24. A CAR IS TO BE RELEASED AT POINT “A.”
WHICH TRACK DESIGN WILL HAVE IT GOING THE FASTEST AT “B?” A B
ALL THE SAME! SINCE THE CHANGE IN HEIGHT IS THE SAME, THE
CHANGE IN POTENTIAL ENERGY IS THE SAME, AND SO IS THE CHANGE IN KINETIC ENERGY

25. B A
WILL THE CAR MAKE IT OVER HILL B, WHICH IS HIGHER THAN HILL A?

26. K K K B P A P P
WILL THE CAR MAKE IT OVER HILL B, WHICH IS HIGHER THAN HILL A?
YES; IT HAS PLENTY OF TOTAL ENERGY.

27. GO BACK TO THE TOP OF LOOP:
FREE-BODY DIAGRAM d
F NET = m a
F NET = m v2/ R
F NET = Fg + FN
Fg + FN = m v2/ R
FN = m v2/ R - Fg
SKETCH FN Fg a
F NET
What you feel is FN ,
the force on your body
If FN = 0, then Fg = m v2top / R. A little more algebra and v2top = Rg

28. GO BACK TO BOTTOM OF LOOP:
A LITTLE ALGEBRA, AND FN = 6 Fg AT THE BOTTOM

29. BIOLOGICAL CONSIDERATIONS
Increased blood pressure
Can rupture weak spots in arteries, etc
Can damage arteries
Brain Damage
6 g’s is the limit for many people; too much for some

30. ENGINEERING CONSIDERATIONS
There is friction – and it varies with wear, lubrication
There is air drag – and it varies with wind
Safety in keeping car on the track
Structural strength of the support
Speed adjustment – compensate for variations

31. MY TWO QUESTIONS:
WHAT PULLS THE TRAIN?
Nothing, after the original lift … Energy
2. WHY NOT CIRCLES FOR THE LOOPS?
Too many g’s …