#mainpage --- ## Velocity basics $\large\color{LightCyan}v$ - Velocity is the **rate of change of position**. The greater the velocity of an object the faster that object is moving. - Although velocity is usually measured in *meters per second $\small \frac{m}{s}$* it can **technically be measured using any unit that relates a unit of distance to a unit of time.** *For example, miles per hour would be considered a velocity measurement.* For information on unit conversion see [[Mathematical relationships]]. - Velocity is a ==**vector quantity**== **which means that it has both a magnitude and a direction**. The magnitude of an object's velocity describes the rate at which its position is changing and the direction represents the direction of that change. - If the velocity of an object **is positive, that indicates it is moving to the right or up**, if it is negative **the object is moving to the left or down** *(In most coordinate systems)*. The greater an object's velocity is **in either direction, the faster that object is moving.** - The **derivative of an object's velocity vs time function is equal to its corresponding acceleration vs time function**. This works because the slope/derivative of a function is equal to its rise over run. Since velocity is equal to - Likewise, the **indefinite** integral of a function relating velocity and time is equal to the displacement of that object over the bounds. $\newline$ $\text{ Velocity: }\space\color{LightCyan}v=\frac{\Delta\space x}{\Delta\space t}\space\space\space\small\color{grey}\text{where }\space\Big(\Delta\space x = x_f-x_i\Big)\space\text{ and }\space\Big(\Delta\space t=t_f-t_i\Big)$ $\newline$ $\text{Velocity from acceleration: }\space \Huge\boxed{\;\large\color{LightCyan}v_f=v_i+at\;}\space\normalsize\text{\color{white} and }\space\Huge\boxed{\large\color{LightCyan}v_f^2=v_i^2+2a\big[\Delta x\big]}$ >[!tip] >##### *Speed vs velocity* >- Speed is a measure of **magnitude only** where as velocity is a measure of **both magnitude and direction.** In other words, speed is a **scalar quantity** while velocity is a **vector quantity**. >- This means that to find speed you must find the absolute value of an object's velocity: >$\Huge\boxed{\,\normalsize\text{speed }=\;|v|\,}$ >[!quote] >##### *Velocity vs time graph* >- **The derivative of a velocity vs time function is equal to that object's corresponding acceleration vs time function.** Likewise, the slope/derivative of an object's velocity vs time graph is equal to it's **instantaneous** acceleration **at that point in time**. The slope/derivative between two points is equal to the **average** acceleration between that period of time. > - **The integral of an object's velocity as a function of time is equal to its DISPLACEMENT as a function of time time. *(not position or total distance traveled!).*** > - To find total distance traveled take the integral of the **absolute value of a function that represents an object's velocity over time.** The bounds of this integral denote the *time between* which the displacement occurs. >- If an object's velocity vs. time graph **crosses the x axis** the object has changed directions. The further away from zero *in either direction* that an object's velocity is the faster that object is going and if the graph is **linear** the object's position is changing either exponentially **faster** or exponentially **slower**. > - If the graph is flat at any x value other then 0 the object's **position is changing linearly**. ></br> > >![[Motionvelocitygraph.png|855]] --- ## *Average vs instantaneous velocity* - The instantaneous velocity of an object is its **velocity at a specific moment in time**. - The slope of a **line that is tangent** to the **position vs time graph of an object** at a specific point is equal to its instantaneous velocity at that specific moment. - Alternatively, you can find an object's instantaneous velocity at a specific point by evaluating the derivative of its **position vs time function** at that point. $\text{Instantaneous velocity: }\;\;\Huge\boxed{\normalsize\color{lightcyan}v_{\small inst}=v'\big(t\big)}$ $\newline$ - The average velocity of an object is its **average velocity between two moments in time**. - Using an object's velocity vs time graph you can **visually "calculate"** an object's average velocity between two points by first, **drawing the shortest linear line that connects those two points on the graph**. The slope of this line will be **equal to the object's average velocity between those two points**. - To mathematically calculate the average velocity of an object between two moments in time we can use the following formula for the slope of this line: $\text{Average velocity: }\;\;\Huge\boxed{\,\normalsize\color{lightcyan}v_{\small avg}=\frac{x_{\small f}-x_{\small i}}{t_{\small f}-t_{\small i}}\;}$ $\newline$ --- ## Velocity and acceleration - Recall that [[Acceleration|acceleration]] is equal to the **rate of change of velocity**. - Since acceleration causes an object's velocity to change over time physicists often need to distinguish between **the initial and final velocity of an object**.# These values are usually denoted as either $\small v_{\small i}$ or $\small v_{\small f}$ and are commonly used in equations that involve both **acceleration and time**. - In most equations $v_{\small f}$ denotes the velocity of an object after some time specified by $\small t$ has passed. In these situations, $v_{\small i}$ denotes the initial velocity of the object when time equals zero. - Through algebraic manipulation we can **solve these equations for $v_{\small i}$ and $v_{\small f}$**. These derived equations are shown below. The variable $\small t$ denotes the amount of time that has passed between the *initial* and *final* velocity, $\small\Delta x$ denotes the distance traveled by the object during this time and $\small a$ obviously denotes the acceleration of the object. $\newline$ $\text{\large Other velocity equations}$ $\begin{aligned}\small\text{Final velocity from initial velocity,}\\\small\text{displacement and acceleration}\end{aligned}\;\Bigg]\;\;\longrightarrow\;\;\Huge\boxed{\,\large\color{lightcyan}v_{\small f}=\sqrt{v_{\small i}^2+2a\,\Delta x}\;}$ $\begin{aligned}\small\text{Initial velocity from final velocity,}\\\small\text{displacement and acceleration}\end{aligned}\;\Bigg]\;\;\longrightarrow\;\;\Huge\boxed{\,\large\color{lightcyan}v_{\small i}=\sqrt{v_{\small f}^2-2a\,\Delta x}\;}$ $\begin{aligned}\small\text{Initial velocity from displacement,}\\\small\text{change in time and acceleration}\end{aligned}\;\Bigg]\;\;\longrightarrow\;\;\Huge\boxed{\large\color{lightcyan}\,v_{\small i}=\frac{\Delta x-\frac{1}{2}at^2}{t}\,}\;\;$ $\newline$ $\small\color{grey}\text{Recall: }\Delta x=x_{\small f}-x_{\small i}$ $\newline$ --- ## Calculator <iframe src="https://app.calconic.com/api/embed/calculator/648bf2ee28c7470029d3f09e" sandbox="allow-same-origin allow-forms allow-scripts allow-top-navigation allow-popups-to-escape-sandbox allow-popups" title="Calconic_ Calculator" name="Calconic_ Calculator" height="3000" scrolling="no" style="width: 100%; border: 0; outline: none;"></iframe>