Normal and Tangential components of Acceleration

v = dR/dt = dR/ds * ds/dt = T * ds/dt
a = dv/dt = dT/dt * ds/dt + T * =
= dT/dt * (ds/dt)^2 + T *
dT/ds = K*N,

a = K(ds/dt)^2 * N + * T
a = aN * N + aT * T
aN = K*(ds/dt)^2 = K |v|^2 = (|v|^2)/

aT = = d|v|/dt
aN

Example: 1
Circle:
for constant speed:

Example 2:
Find the distance (d) from a point (S), to a line (PB).

u is the unit vector in the direction of the line.


Example 3:
Find the distance between the point S(1,1,1), and the line: x = -t, y = 1 + t, and z = t
v = -i + j + k
u = (-i + j + k)/3^(1/2)
Chose t = 0
P = (0,1,0), => PS = i + k
=> d = 2^(1/2)