HTSEFP: Motion along curves

1The works¶

1.1Problem¶

Given parametric equations for the motion of a particle, obtain "the works"

1.2General solution¶

Velocity vector: Differentiate the position vector $\vec R = x \hat i + y \hat j + z\hat k$

Speed: magnitude of velocity (length of velocity vector)

Unit tangent vector: Divide the velocity vector by its length

Acceleration vector: Differentiate the velocity vector

Kappa: $\frac{\dot{x} \ddot{y} - \dot{y}\ddot{x}}{(\dot{x}+\dot{y})^{3/2}}$ or $\displaystyle \frac{\lVert \vec a_N \rVert}{\lVert \vec v \rVert^2} = \frac{\sqrt{\lVert \vec a \rVert^2 - a_T^2}}{\lVert \vec v \rVert^2}$

Radius: Inverse of kappa

Tangential component of acceleration: derivative of the velocity (magnitude?)

Normal component of acceleration: $\vec a_N = \sqrt{\lVert \vec a \rVert^2 - \lVert a_T \rVert^2}$

Unit normal vector: $\vec a = \vec a_N \hat N + \vec a_T \hat T$ so get it from that

Unit binormal vector: cross-product of unit normal and unit tangent

Radial and circumferential components of velocity and acceleration: Convert to polar coordinates, use dot products. $\mu_r = \cos\theta \hat i + \sin\theta\hat j; \,\mu_{\theta} = -\sin\theta\hat i + \cos\theta\hat j$

1.3Examples¶

• Assignment 5, question 12