Final answer:
To find T and N, we first normalize the velocity vector. T is the normalized velocity vector, and N is the cross product of T and the second derivative of the position vector. The tangential component of acceleration, aT, is the projection of the second derivative onto T, and the normal component of acceleration, aN, is the projection of the second derivative onto N.
Step-by-step explanation:
In order to find the tangential and normal components of acceleration, we first need to find the values of T and N. We are given the position vector function r(t) = (t, 6t^2, t^3) and the value of t at which we need to find T and N is t=1.
To find T, we need to normalize the velocity vector. The velocity vector v(t) is found by taking the first derivative of r(t) with respect to t. So, v(t) = (1, 12t, 3t^2). Evaluating v(t) at t=1, we get v(1) = (1, 12, 3).
T is found by dividing v(1) by its magnitude. |v(1)| = √(1^2 + 12^2 + 3^2) = √(1 + 144 + 9) = √154. So, T = v(1) / |v(1)| = (1/√154, 12/√154, 3/√154)
Once we find T, N can be found by taking the cross product of T and the second derivative of r(t) with respect to t. The second derivative of r(t) is a(t) = (0, 12, 6t). Evaluating a(t) at t=1, we get a(1) = (0, 12, 6).
N is found by taking the cross product of T and a(1). N = T x a(1) = ((12/√154) * 6, -(3/√154) * 0, -(1/√154) * 12) = (72/√154, 0, -12/√154)
Now, we can find the tangential and normal components of acceleration. The tangential component aT is the projection of a(1) onto T, which is a scalar multiple of T. So, aT = (a(1) · T) * T, where · denotes the dot product. Evaluating a(1) · T, we get a(1) · T = 0 * (12/√154) + 12 * (12/√154) + 6 * (3/√154) = 144/√154.
Therefore, aT = (144/√154) * (1/√154, 12/√154, 3/√154) = (144/154, 1728/154, 432/154).
The normal component aN is the projection of a(1) onto N, which is a scalar multiple of N. So, aN = (a(1) · N) * N. Evaluating a(1) · N, we get a(1) · N = 0 * (72/√154) + 12 * 0 + 6 * (-12/√154) = -72/√154.
Therefore, aN = (-72/√154) * (72/√154, 0, -12/√154) = (-72/154, 0, 144/154).