Kinematics of a Particle in Space

In a Nut Shell: The kinematics of a particle traveling in 3˗D is much more complicated than

in a plane. In addition to unit vectors in the tangential, e_t, and normal, e_n, directions an additional

unit vector, the binormal unit vector, e_b, enters into the description of the motion. See the

figure below.

Kinematics of a Particle in Space, C

r = x i + y j + z k = position vector of particle at point P at any time t

here i is the unit vector in the x-direction , j is the unit vector in the y-direction,

and k is the unit vector in the z-direction,

Definition of terms:

e_t ˗ is the unit tangential vector (tangent to space curve)

e_n ˗ is the unit normal vector (points to center of curvature)

e_b ˗ is the unit binormal vector (normal to both e_t and e_n)

Orthogonal Relations: e_b = e_t x e_n , e_t = e_n x e_b , e_n = e_b x e_t

Osculating plane: contains e_t and e_n Normal plane: contains e_n and e_b

Rectifying plane: contains e_b and e_t

Calculation of Unit Vectors:

e_t = ( dr/dt ) / | dr/dt | , e_n = ( d e_t /dt ) / | d e_t /dt | , e_b = [ dr/dt x d²r/dt² ] / | dr/dt x d²r/dt² |

Calculation of radius of curvature (space curve):

ρ = | dr/dt |³ / | dr/dt x d²r/dt² |

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Copyright © 2019 Richard C. Coddington
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