What is the radius rho of the osculating circle at any time

Question

(a) Consider the curve whose parameterization, for t ≥ 0, is r^→ (t) = , where

x(t) = ₀∫^t sin (u²/2) du

y(t) = ₀∫^t cos (u²/2) du

z(t) = 0

Perform the decomposition a^→ = a_T T^→ +a_N N^→ at any time t ≥ 0.

(b) What is the radius, ρ, of the osculating circle at any time t > 0, for the curve in part (a)? Given that the curve is defined for all t ≥ 0, why does this question not ask for the radius, at any time t ≥ 0?

(c) The magnitude of the vector N^m→ (t) is considered to be a useful estimator of the discomfort which a human will experience while riding in a train, which is rounding a track parameterized by r^→ (t). What is the magnitude of this vector, at all times t ≥ 0, for the curve in part (a)?

(d) What is the arc length, s(t), at any time t > 0, for the curve in part (a)? Now give the arc length parameterization, r^→ (t(s)) = r^→ (s), for the curve.