Kinematic’s

Symbols

  • \$V\$ or \$V_{f}\$ → Final velocity

  • \$a\$ or \$\vec{a}\$ → Acceleration

  • \$t\$ → Time

  • \$V_{0}\$ → Initial Velocity

  • \$\overline{S}\$ → Average Speed

  • \$X_{F}\$ → Final Position

  • \$\Deltax\$ → Displacement of x axis

  • \$\Deltay\$ → Displacement of y axis

  • \$\vec{V_{a}}\$ → Average Velocity

  • \$S_{i}\$ → Instantaneos Speed

  • \$V_{i}\$ → Instantaneos Velocity

  • \$X_{0}\$ → Initial Position

  • \$V_{t}\$ → Terminal Velocity

  • \$\vec{S}\$ → Displacement

  • \$\vec{V}\$ → Vector

  • \$S_{a}\$ → Speed Average

  • \$w\$ → weight

Equations

Basic

  • \$V = V_{0}+at\$

  • \$(\Deltax or \Deltay) = \frac{V + V_{0}}{2}t\$

  • \$(\Deltax or \Deltay) = V_{0}t + 1/2at^2\$

  • \$V^2 = V_{0}^2 + 2a\Deltax\$

Extra

  • \$\vec{V} = \vec{S}/t\$ or \$r = d/t\$

  • \$\vec{V_{a}} = \frac{\Deltax}{t}\$

  • \$\vec{a} = \frac{\Delta\vec{V}}{t}\$

  • \$a = \frac{V_{f} - V_{i}}{\Deltat}\$

  • \$\vec{V_{a}} = \frac{V_{f} + V_{i}}{2}\$ ???

  • \$\vec{S} = (\vec{V_{a}})\Deltat\$

  • \$\vec{S} = \frac{V_{f}^2-V_{i}^2}{2\vec{a}}\$

  • \$2\vec{a}\vec{s} = V_{f}^2 - V_{i}^2\$

  • \$\DeltaV = a\Deltat\$

  • \$\frac{\Deltaa}{\Deltat} = jerk\$

  • \$V_{i} = V_{f} - a\Deltat\$

  • \$\Deltat = \frac{V_{f} - V_{i}}{a}\$

  • \$S_{a} = \frac{\text{distance}}{\Deltat}\$

If the divisor is t then it is assumed to be \$\Deltat\$
kinematic formulas are only accurate if the acceleration is constant during the time interval considered

2D Projectile Motion

There’s no acceleration in the horizontal direction since gravity does not pull projectiles sideways

  • \$Delta x=v_xt \quad\$

An object can start with a horizontal component of velocity, yet have zero vertical component of velocity.

Instentaneos Rate of Change

\$\lim_{\Delta \to 0} \frac{f(x + \Deltax) - f(c)}{\Deltax} = f'(x)\$ \$or\$ \$\lim_{x \to c} \frac{f(x) - f(c)}{x - c} = f'(c)\$
\$f'(x) = v(t) -> \text{Instentaneos Velocity}\$ \$f''(x) = a(t) -> \text{acceleration}\$ \$|f'(x)| = s(t) -> \text{Instentaneos speed}\$ \$f'''(x) -> jerk\$

Force

Symbols

  • \$\vec{F}\$ → Force of Gravity

  • \$g\$ → Acceleration of Gravity

  • \$m\$ → mass of object

  • \$r\$ → distance between objects

  • \$\vec{a_{g}}\$ → acceleration due to gravity

Equations

  • \$\vec{F} = m \cdot g\$

  • \$\vec{F} = m \cdot a\$

  • \$a = \vec{F}/m\$

  • \$a = g\$

  • \$\vec{F} = G \frac{m_{1}m_{2}}{r^2}\$

  • \$w = \frac{m}{g}\$

Gravity

\$g = G \frac{m}{r^2} = 9.81 frac{m}{s^2}\$

  • \$m\$ → Mass of the Earth

  • \$r\$ → Radius of the Earth

Temrinal Velocity

\$V_{t} = frac{sqrt{2mg}}{pAC_{d}}\$

  • \$V_{t}\$ represents terminal velocity

  • \$m\$ is the mass of the falling object

  • \$g\$ is accleration due to gravity

  • \$C_{d}\$ is the drag coefficent

  • \$p\$ is the density through which th object is falling

  • \$A\$ is the projected area of the object

Graphing

Velocity vs. Time Graphs

  • Slope is acceleration and a way to find instentaneos velocity

  • Area is displacement

Displacement

Acceleration vs. Time Graphs

  • Slope is jerk

  • Area is change in velocity

  • Typical y-axis: \$m/s^2\$

Change In Velocity

Using Quadradic Formula

\$\Delta y=v_{0y} t+\dfrac{1}{2}a_yt^2 \quad \text{(Start with the third kinematic formula.)}\$
\$12.2\text{ m}=(18.3\text{ m/s})t+\dfrac{1}{2}(-9.81\dfrac{\text{ m}}{\text{ s}^2})t^2 \quad \text{(Plug in known values.)}\$
\$0=\dfrac{1}{2}(-9.81\dfrac{\text{ m}}{\text{ s}^2})t^2+(18.3\text{ m/s})t -12.2\text{ m} \quad \text{(Put it into the form of the quadratic equation.)}\$
\$a=\dfrac{1}{2}(-9.81\dfrac{\text{ m}}{\text{ s}^2}) \$ \$b=18.3\text{ m/s} \$ \$c=-12.2\text{ m}\$
\$t=\dfrac{-18.3\text{ m/s}\pm\sqrt{(18.3\text{ m/s})^2-4[\dfrac{1}{2}(-9.81\dfrac{\text{ m}}{\text{ s}^2})(-12.2\text{ m})]}}{2[\dfrac{1}{2}(-9.81\dfrac{\text{ m}}{\text{ s}^2})]}\$
\$t=0.869\text{s}\ \text{and}\ t=2.86\text{s}\$

Links

https://phys.libretexts.org/Bookshelves/University_Physics/University_Physics_(OpenStax)/Book%3A_University_Physics_I_-_Mechanics_Sound_Oscillations_and_Waves_(OpenStax)/11%3A__Angular_Momentum/11.02%3A_Rolling_Motion