Physics 1600 4 Kinematics in three dimensions

From Physics 1600 Review Plan

• Physics 1600 4 Kinematics in three dimensions 4 Kinematics in three dimensions 95 4.1 Vectors … … … … … … … … … … . . 95 4.1.1 Basic definitions and notation, inner product … … 95 4.1.2 Cross product … … … … … … … … 99 4.1.3 Basis vectors … … … … … … … … . 101 4.1.4 Vectors in two dimensions … … … … … . . 104
• Is writing V=|V| Vhat the first step in polar coordinates?
• Finding velocity
  • $\dot{\hat{V}}$ is complicated, so it is further decomposed into direction and magnitude]
    • magnitude $|\dot{\hat{V}}|\approx \dot{\theta }$ can be derived
    • direction: $\hat{k}\times \hat{V}$
      • Where does the definition of k come from?
    • As a result, $\dot{\hat{V}}=\dot{\theta }\hat{\theta }$
• Finding acceleration
  • $\hat{\dot{\theta }}$ is complicated, so it is decomposed into direction and magnitude as usual
    • magnitude: $\dot{\theta }$
    • direction: $\hat{k}\times \hat{\theta }=-\hat{r}$

4.1.5 Vector derivatives…105

• $[\hat{V}]$=1, indeed
• I want to be able to explain this calculation by drawing my own diagrams, etc.
  • $\dot{\theta }(\hat{n}\times \hat{V})$, both n and V are unit vectors, so their product is also a unit vector
  • I also want to develop an intuition for the direction of the normal vector 4.1.6 Vector derivatives, reprise …109 4.2 Locations and position vectors…112 4.3 Velocity and acceleration vectors …114 4.4 Speed, distance, Euclidean metric…117 4.5 Separable systems in Cartesian coordinates … … … . . 119
• Verification: What does it mean for a system to be non-separable? (blu3mo)
  • https://math.libretexts.org/Courses/Monroe_Community_College/MTH_211_Calculus_II/Chapter_8%3A_Introduction_to_Differential_Equations/8.3%3A_Separable_Differential_Equations
  • If a_x(x,t)=~~~, then the equation cannot be solved when both x and t are involved?
  • TODO: Confirm with Rec (blu3mo) 4.6 Motion in a plane using polar coordinates … … … … 121
• Definition of $\vec{a}$
  • I am starting to understand why there is a Θ component in $\hat{r}$
    • The content of $\hat{r}(..)$ does not mean the acceleration of r, but the sum of accelerations in the r direction
    • So, if r were constant, we would need to apply an acceleration to maintain circular motion 4.7 Coordinate Transformations …126 4.7.1 Translations …126 4.7.2 Rotations…127