Summary of Motion in Space

If [latex]{\bf{r}}(t)[/latex] represents the position of an object at time [latex]t[/latex], then [latex]{\bf{r}}^{\prime}(t)[/latex] represents the velocity and [latex]{\bf{r}}^{\prime\prime}(t)[/latex] represents the acceleration of the object at time [latex]t[/latex]. The magnitude of the velocity vector is speed.
The acceleration vector always points toward the concave side of the curve defined by [latex]{\bf{r}}(t)[/latex]. The tangential and normal components of acceleration [latex]a_{\bf{T}}[/latex] and [latex]a_{\bf{N}}[/latex] are the projections of the acceleration vector onto the unit tangent and unit normal vectors to the curve.
Kepler’s three laws of planetary motion describe the motion of objects in orbit around the Sun. His third law can be modified to describe motion of objects in orbit around other celestial objects as well.
Newton was able to use his law of universal gravitation in conjunction with his second law of motion and calculus to prove Kepler’s three laws.

Key Equations

Velocity
[latex]{\bf{v}}(t)={\bf{r}}^{\prime}(t)[/latex]
Acceleration
[latex]{\bf{a}}(t)={\bf{v}}^{\prime}(t)={\bf{r}}^{\prime\prime}(t)[/latex]
Speed
[latex]v(t)=\parallel{\bf{v}}(t)\parallel=\parallel{\bf{r}}^{\prime}(t)\parallel=\frac{ds}{dt}[/latex]
Tangential component of acceleration
[latex]a_{\bf{T}}={\bf{a}}\cdot{\bf{T}}=\frac{\bf{v}\cdot\bf{a}}{\parallel{\bf{v}}\parallel}[/latex]

Normal component of acceleration
[latex]a_{\bf{N}}={\bf{a}}\cdot{\bf{N}}=\frac{\parallel{\bf{v}}\times{\bf{a}}\parallel}{\parallel{\bf{v}}\parallel}=\sqrt{\parallel{\bf{a}}\parallel^{2}-{a_{\bf{T}}}^{2}}[/latex]

Kepler’s laws of planetary motion: three laws governing the motion of planets, asteroids, and comets in orbit around the Sun

normal component of acceleration: the coefficient of the unit normal vector [latex]{\bf{N}}[/latex] when the acceleration vector is written as a linear combination of [latex]{\bf{T}}[/latex] and [latex]{\bf{N}}[/latex]

projectile motion: motion of an object with an initial velocity but no force acting on it other than gravity

tangential component of acceleration: the coefficient of the unit tangent vector [latex]{\bf{T}}[/latex] when the acceleration vector is written as a linear combination of [latex]{\bf{T}}[/latex] and [latex]{\bf{N}}[/latex]