# Kinematics & Dynamics ## Position, Velocity, and Acceleration > [!example]- The Car Example > kachow > ![[cars cars baby.gif|225]] > > The *position* function describes the car’s location. > > The *velocity* function is what is displayed on the car’s speedometer. > > The *acceleration* is how hard the driver is pressing on the gas pedal (if accelerating) or the breaks (if decelerating). > > --- > **Caveats:** > - It’s still important to distinguish speed and velocity here: > - If the car was going forward, then both would be the same. > - However, if the car was going in reverse, the velocity would now be negative and the speed would be its absolute value. ### Mathematical Representations > See also: > - [[Scalar Quantities]] ![[Pasted image 20250113115201.png|376]] ![[Calculus Loops.png|375]] ### “Instantaneous” Values > See also: > - [[Rates of Change]] ## Position ## Velocity (& Speed) Average velocity is the displacement divided by the time interval. $v_{avg}$ The **instantaneous velocity** of an object is the limit of the average velocity as the elapsed time [[Limits|approaches zero (limits)]], otherwise known as the derivative of $x$ with respect to $t$. - [?] What does it mean when velocity “changes direction”? ## General Concepts ### Displacement vs Distance ![[Pasted image 20250117203446.png|350]] Displacement can be defined as the sum of the individual displacements $\Delta x_{\text{Total}} = \sum \Delta x_{i}$ ## One-Dimensional Motion ### Velocity and Speed > See also: > - [[Position, Velocity, and Acceleration]] | Definition | Equation | | --------------------------------------- | ----------------------------------------------------------------- | | Average velocity | $\bar{v} = \frac{v_0+v}{2}$ | | Average velocity (for const $a$) | $\bar{v}=\frac{\Delta x}{\Delta t} = \frac{x_2-x_1}{t_2 - t_1}$ | | Position from average velocity | $x=x_0 + \bar{v}t$ | | Velocity from acceleration | $v=v_{0}-at$ | | Position from velocity and acceleration | $x=x_{0}+v_{0}t-\frac{1}{2}at^2$ | | Velocity from distance | $v^2=v^2_{0}-2a(x-x_{0})$ | ## Free-Fall Motion https://www.youtube.com/watch?v=g76OWKZdxpg **Free-fall motion** is defined as motion that is *only under the influence of gravity* and no other forces. We often ignore air resistance as a factor to simplify calculations. https://wiki.guildwars2.com/wiki/Tempest