Ch. 4 Key Equations - University Physics Volume 1

Key Equations

Position vector	$\vec{r} (t) = x (t) \hat{i} + y (t) \hat{j} + z (t) \hat{k}$
Displacement vector	$Δ \vec{r} = \vec{r} (t_{2}) - \vec{r} (t_{1})$
Velocity vector	$\vec{v} (t) = lim_{Δ t \to 0} \frac{\vec{r} (t + Δ t) - \vec{r} (t)}{Δ t} = \frac{d \vec{r}}{d t}$
Velocity in terms of components	$\vec{v} (t) = v_{x} (t) \hat{i} + v_{y} (t) \hat{j} + v_{z} (t) \hat{k}$
Velocity components	$v_{x} (t) = \frac{d x (t)}{d t} v_{y} (t) = \frac{d y (t)}{d t} v_{z} (t) = \frac{d z (t)}{d t}$
Average velocity	${\vec{v}}_{avg} = \frac{\vec{r} (t_{2}) - \vec{r} (t_{1})}{t_{2} - t_{1}}$
Instantaneous acceleration	$\vec{a} (t) = lim_{t \to 0} \frac{\vec{v} (t + Δ t) - \vec{v} (t)}{Δ t} = \frac{d \vec{v} (t)}{d t}$
Instantaneous acceleration, component form	$\vec{a} (t) = \frac{d v_{x} (t)}{d t} \hat{i} + \frac{d v_{y} (t)}{d t} \hat{j} + \frac{d v_{z} (t)}{d t} \hat{k}$
Instantaneous acceleration as second derivatives of position	$\vec{a} (t) = \frac{d^{2} x (t)}{d t^{2}} \hat{i} + \frac{d^{2} y (t)}{d t^{2}} \hat{j} + \frac{d^{2} z (t)}{d t^{2}} \hat{k}$
Time of flight	$T_{tof} = \frac{2 (v_{0} sin θ_{0})}{g}$
Trajectory	$y = (tan θ_{0}) x - [\frac{g}{2 {(v_{0} cos θ_{0})}^{2}}] x^{2}$
Range	$R = \frac{v_{0}^{2} sin 2 θ_{0}}{g}$
Centripetal acceleration	$a_{C} = \frac{v^{2}}{r}$
Position vector, uniform circular motion	$\vec{r} (t) = A cos ω t \hat{i} + A sin ω t \hat{j}$
Velocity vector, uniform circular motion	$\vec{v} (t) = \frac{d \vec{r} (t)}{d t} = − A ω sin ω t \hat{i} + A ω cos ω t \hat{j}$
Acceleration vector, uniform circular motion	$\vec{a} (t) = \frac{d \vec{v} (t)}{d t} = − A ω^{2} cos ω t \hat{i} - A ω^{2} sin ω t \hat{j}$
Tangential acceleration	$a_{T} = \frac{d \| \vec{v} \|}{d t}$
Total acceleration	$\vec{a} = {\vec{a}}_{C} + {\vec{a}}_{T}$
Position vector in frame S is the position vector in frame $S^{'}$ plus the vector from the origin of S to the origin of $S^{'}$	${\vec{r}}_{P S} = {\vec{r}}_{P S^{'}} + {\vec{r}}_{S^{'} S}$
Relative velocity equation connecting two reference frames	${\vec{v}}_{P S} = {\vec{v}}_{P S^{'}} + {\vec{v}}_{S^{'} S}$
Relative velocity equation connecting more than two reference frames	${\vec{v}}_{P C} = {\vec{v}}_{P A} + {\vec{v}}_{A B} + {\vec{v}}_{B C}$
Relative acceleration equation	${\vec{a}}_{P S} = {\vec{a}}_{P S^{'}} + {\vec{a}}_{S^{'} S}$