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Chapter8Motion

Objectives

This unit covers the following ideas.

  1. Develop formulas for the velocity and position of a projectile, if we neglect air resistance and consider only acceleration due to gravity. Show how to find the range, maximum height, and flight time of the projectile.

  2. Develop the \(TNB\) frame for describing motion. Explain why \(\vec T\text{,}\) \(\vec N\text{,}\) and \(\vec B\) are all orthogonal unit vectors, and how to perform the computations to find these three vectors.

    • Compute the unit tangent and unit normal vector of space curves

  3. Explain the concepts of curvature \(\kappa\text{,}\) radius of curvature \(\rho\text{,}\) center of curvature, and torsion \(\tau\text{.}\) Including what these quantities mean geometrically.

  4. Find the tangential and normal components of acceleration. Show how to obtain the formulas \(a_T=\frac{d}{dt}|\vec v|\) and \(a_N=\kappa |\vec v|^2=\frac{|\vec v|^2}{\rho}\text{,}\) and explain what these equations physically imply.

Table 1 summarizes most of the concepts we'll discuss. The goal of this chapter is to explain how the vectors in this table are related. You'll also find this Sage notebook (click on the link) can greatly speed up all the computations in this chapter. Prof. Woodruff has also created a YouTube playlist to go along with this section. There are 11 videos, each 4-6 minutes long.

Quantity Symbol Formula
Position (“r”adial vector) \(\vec r\) \(\vec r(t) = (x(t),y(t),z(t))\)
Velocity \(\vec v\) \(\ds \vec v(t) = \frac{d\vec r}{dt}\)
Speed \(v = \ds\frac{ds}{dt}\) \(\ds v(t) = |\vec v(t)|\)
Acceleration \(\ds \vec a\) \(\ds \vec a(t) = \frac{d \vec v}{dt}= \frac{d^2\vec r}{dt^2}= \frac{d}{dt}\frac{d\vec r}{dt}\)
Unit Tangent Vector \(\vec T\) \(\ds\frac{d\vec r}{ds} = \frac{d\vec r/dt}{ds/dt} = \frac{\vec r^\prime(t)}{|\vec r^\prime(t)|}\)
Curvature Vector \(\vec \kappa\) \(\ds\frac{d\vec T}{ds} =\frac{d\vec T/dt}{ds/dt} = \frac{d\vec T/dt}{|\vec v|} = \frac{\vec T^\prime(t)}{|\vec r^\prime(t)|}\)
Curvature (a scalar) \(\kappa\) \(\ds \left|\frac{d\vec T}{ds}\right| =\left|\frac{d\vec T/dt}{ds/dt}\right| = \frac{\left|d\vec T/dt\right|}{|\vec v|}= \frac{|\vec T^\prime(t)|}{|\vec r^\prime(t)|}\)
Curvature of \(y=f(x)\) \(\kappa(x)\) \(\ds \kappa(x) = \frac{|f''(x)|}{(1+(f')^2)^{3/2}}.\)
Principal unit normal vector \(\vec N\) \(\ds \frac{d\vec T/dt}{|d\vec T/dt|} = \frac{\vec T^\prime(t)}{|\vec T^\prime(t)|}=\frac{1}{\kappa}\frac{d\vec T}{ds} = \frac{1}{\kappa |\vec v|}\frac{d\vec T}{dt}\)
Binormal vector \(\vec B\) \(\vec T\times\vec N\)
Radius of curvature \(\rho\) \(1/\kappa\)
Center of curvature \(\vec r(t)+\rho(t)\vec N(t)\)
Torsion \(\tau\) \(\ds \pm\left|\frac{d\vec B}{ds}\right|\) (pick the sign) or \(\ds-\frac{d\vec B}{ds}\cdot \vec N\)
Tangential Component of acceleration \(a_T\) \(\ds \vec a \cdot \vec T = \frac{d}{dt}|\vec v|\)
Normal Component of acceleration \(a_N\) \(\ds \vec a \cdot \vec N = \kappa \left(\frac{ds}{dt}\right)^2 = \kappa |\vec v|^2\)
Acceleration (sum the components) \(\vec a\) \(\vec a = a_T\vec T+a_N\vec N = \left(\frac{d}{dt}|\vec v|\right) \vec T +\left(\kappa |\vec v|^2\right) \vec N\)
Table8.0.1This table summarizes the key ideas in this unit. Most of our work in this unit will be to explain the connections between these variables.