Normal and Tangential Components

Question

Find the tangential and normal components of the acceleration vector for the curve →r(t)= ⟨4t,3t^2,4t^4⟩ at the point t=2

Answer format: a(2) = (Decimal #) T + (Decimal #) N

(Give your answer to two decimal places.

Answer 1

Probably the fastest way is to find a(t) = d²r/dt²

Then a_t = a•v/v The dot product of acc. vector in tangent direction (unit tangent vector is v/v

a_n = |a x v|/v The cross product with tangent vector gives normal component of acc. vector.

v = dr/dt = <4,6t,16t³> |v| = v = sqrt(16 + 36t² + 256t⁶) a = <0,6,48t²>

a_t = (36t + 768t⁵)/v

a_n = |96t³ - 288t³, 192t²,-24|/v

Plug in 2 for t. You can calculate v as a number and the magnitude is the sqrt of sum of the squares of the vector in the a_n equation.

You can also find the curvature,κ, and use v' = a_t and κv² = a_n