# Physics equations/Kinematics

#### "D" or delta as difference

${\displaystyle {\rm {d,\delta ,\Delta }}}$, and ${\displaystyle \partial }$ all mean difference  : ${\displaystyle \Delta x=x_{\text{final}}-x_{\text{initial}}}$   , ${\displaystyle \Delta t=t_{\text{final}}-t_{\text{initial}}}$.   Other subscripts are also used (e.g., ${\displaystyle t_{\text{f}}-t_{\text{final}}}$, or ${\displaystyle t_{2}-t_{1}}$, or ${\displaystyle t-t_{0}}$.)

#### One dimensional velocity and acceleration

Velocity is the rate at which position changes. Acceleration is the rate at which velocity changes. If the time interval is not infinitesimally small, we refer to these as "average" rates. The average velocity or acceleration is often denoted by a bar above. Alternatives to ${\displaystyle {\bar {v}}}$ to are the brakcet ${\displaystyle }$ and the subscript ${\displaystyle v_{\text{ave}}}$.

${\displaystyle {\bar {v}}={\frac {\Delta x}{\Delta t}}={\frac {x_{f}-x_{i}}{t_{f}-t_{i}}}}$,   ${\displaystyle {\bar {a}}={\frac {\Delta v}{\Delta t}}={\frac {v_{f}-v_{i}}{t_{f}-t_{i}}}}$.

Instantaneous velocity and acceleration are derivatives: ${\displaystyle v(t)=dx/dt}$,   ${\displaystyle a(t)=dv/dt=d^{2}x/dt^{2}}$

#### Uniformly accelerated motion in one and two dimensions

{\displaystyle {\begin{aligned}x=x_{0}+v_{0}t+{\frac {1}{2}}at^{2}&\rightarrow {\vec {r}}={\vec {r}}_{0}+{\vec {v}}_{0}t+{\frac {1}{2}}{\vec {a}}t^{2}\\v=v_{0}+at&\rightarrow {\vec {v}}={\vec {v}}_{0}+{\vec {a}}t\\&\rightarrow v^{2}=v_{0}^{2}+{\vec {a}}\cdot ({\vec {r}}-{\vec {r}}_{0}).\end{aligned}}}

The student should first master this concept in the simple notation before attempting to generalization that follows the ${\displaystyle \rightarrow }$ symbol. We shall employ this symbol to highlight how physics is often expressed using sequence of simple-to-powerful notation (click the link for further discussion).

Free-fall occurs when gravity is the only force that acts. The vector acceleration is ${\displaystyle {\vec {a}}=-g{\hat {j}}}$, where ${\displaystyle g\approx 9.8m/s^{2}}$ is the acceleration of gravity at Earth's surface. The particle is at, ${\displaystyle {\vec {r}}}$ ${\displaystyle =}$${\displaystyle x{\hat {i}}+y{\hat {j}}}$, and the velocity vector,${\displaystyle {\vec {v}}}$, is usually written in component form as,${\displaystyle v_{x}=v\cos \theta }$, and ${\displaystyle v_{y}=v\sin \theta }$. The trajectory of free fall can be obtained from the initial conditions,