Roller coaster change in acceleration?

[Arkane1337]

Homework Statement

You have taken a job as senior
hench-person with Dr. Caasi Notwen, arch-nemesis of
James Bond. As with all of Bond’s enemies, Dr. Notwen
would like to develop an overly complicated method of
dispatching the British super-spy, so he has tasked you
with designing a “roller coaster of death.” This
frictionless roller coaster is to be designed such that
when the rider (James Bond) reaches the nadir (point B)
on the track, the acceleration he experiences will exceed
the limit of human survival (10 g’s where g = 9.8 m/s²).
At point B the track has a radius of curvature of 25.0 m..

(a) From what height h (Point A) must the cart start (from rest) to ensure Bond’s demise?

Homework Equations

a_c=v²/R
PE=mgh
KE=(1/2)mv²

The Attempt at a Solution

10g's = 98.0 m/s² ≥ a_c = v²/R

v = sqrt{a_cR} = $\sqrt{98*25}$ = 49.5 m/s


PE = KE
mgh = (1/2)mv²
h = v²/2g = 49.5²/(2*9.8) = 125.0m

*** I'm rather rusty on my centripetal concepts and am wary of my answer; does it look correct to all of you? ***

Thanks in advance!

 

[TSny]

There could be some ambiguity in the wording of the question. Your answer is correct if you want the actual acceleration of Bond to be 10 times g.

However, to "experience 10 g's" could be interpreted as experiencing an apparent weight of 10 times your normal weight. That would mean that you would want the normal force acting on Bond to be 10 times his weight. Or, to put it another way, even if bond just sat at rest at point B he would be experiencing 1 g. So, the effect of the motion should be to add another 9 g's if you wanted Bond to experience a normal force of 10 times his weight.

Not sure of the intended interpretation.

 

[Arkane1337]

There seems to be a second part to the problem, which I'm rather confused on as well; could someone provide any hints?

(b) What force will Bond feel (presuming he survives past Point B) at Point C where the radius of
curvature has been reduced to 15.0 m and is 30 m above point B (i.e., what is the normal force at C)?

 

[Arkane1337]

Anyone?

 

[Nickie]

I would approach this problem by first considering the physical limitations of the human body. The average human body can withstand acceleration up to 4-5 g's before experiencing discomfort and up to 10 g's before losing consciousness. Therefore, designing a roller coaster with an acceleration of 10 g's at the nadir would most likely result in Bond's demise.

To determine the height h at point A, we can use the equation ac = v^2/R, where ac is the centripetal acceleration, v is the velocity, and R is the radius of curvature. We know that ac = 98 m/s^2 and R = 25 m, so we can solve for v.

v = sqrt(ac*R) = sqrt(98*25) = 49.5 m/s

Next, we can use the conservation of energy principle to determine the height h. At point A, Bond has only potential energy, which will be converted to kinetic energy as he moves down the track. The total energy at point A must be equal to the total energy at point B.

PE at point A = mgh = KE at point B = (1/2)mv^2

Solving for h, we get:

h = v^2/2g = (49.5 m/s)^2/(2*9.8 m/s^2) = 127.6 m

Therefore, the cart must start at a height of 127.6 meters (point A) to ensure Bond's demise at point B. However, as a scientist, I would also consider the design and construction of the roller coaster to ensure that it is safe for other riders and that all safety precautions are in place.