For the ride to be comfortable, the magnitude of acceleration must not exceed 32 m/s2. What is the fastest constant speed that a coaster can travel along this section of the track subject to this condition

the maximum possible constant speed is  8 m/sec  

Explanation:

from the image, Given that

r(t) = (2t, t²,t²/3), -5 ≤ t ≤ 5  

Given that the curvature K(t)  = 2 / ( t² + 2)²  

note that t² + 2 ≥ 2  

(t² + 2)² ≥ 4

1 / (t² + 2)² ≤ 1/4

2 / (t² + 2)² ≤ 1/2

Also note that k(0) = 1/2

The normal component of acceleration satisfies aN = kv²

where v = ║v(t)║is the speed of the roller coaster.

The maximum possible normal component of acceleration is 32

so, aN ≤ 32 every where on the track

aN = kv² ≤  1/2v² ≤ 32

v² ≤ 64

Therefore, the maximum possible constant speed is  8 m/sec