--- ## Displacement $\color{LightCyan}\Delta x$ - In physics the term **displacement** has a very specific meaning. **Displacement is a measure your distance from the starting position**, *measured in a straight line along the shortest path from you to the starting position*. Displacement does not **tell you anything about how far an object has traveled**, it tells you only about how **far its current position is from the place where it started**. - Displacement is a **a vector value *(a measure of magnitude* and *direction)***. - Displacement is denoted using the variable: $\small\Delta x$. - The general equation for displacement is: $\small\Delta x = |\text{ current position }-\text{ starting position }|$ - The ==**displacement of an object is equal to the absolute value of the area under its velocity vs time curve**==. - The terms *"direction"* and *"distance"* are distinct from displacement. *"Direction"* usually refers to an object's **position relative to some other place** and *"distance"* usually refers to an object's **distance traveled** which is also different from displacement *(more on total distance later).* - *"Direction" can also be though of as the direction/angle portion of an object's displacement vector.* ##### Calculating displacement - The easiest way to find an object's displacement is use the **Pythagorean theorem to find the straightest line from its starting position to its current position**. - Since an object's displacement is equal to the area under its **velocity vs time function** we can alternately use these **pre integrated equations** to estimate the object's displacement over some period of time. - If we have the graph of an object's velocity vs time we could find that object's displacement by finding the area under the graph over some specified bounds. - These *"pre integrated equations"* shown below are commonly called **"kinematics equations"** and are used all the time to find an object's position after some time **if that object is has a non-zero velocity or acceleration**. *Note: these equations are only truly accurate if the velocity/acceleration is constant on the bounds (time) used.* $\newline$ $\text{Displacement: }\space\Huge\boxed{\color{LightCyan}\large\Delta \vec{x}=\vec{x_f}-\vec{x_i}}$ $\text{Displacement from velocity: }\;\Huge\boxed{\large\space\color{LightCyan}\Delta x=vt\space\text{\normalsize\color{grey} or }\space\frac{1}{2}\Big(v_i+v_f)\cdot t\;}$ $\text{Displacement from velocity and acceleration: }\;\;\Huge\boxed{\color{LightCyan}\large\space \Delta x=vt\space +\space\frac{1}{2}at^2\;}$ $\newline$ >[!danger] >##### *Difference between scalar and vector quantity* >- A scaler quantity refers to a value that only contains **information about magnitude**. >- A vector quantity refers to a value that instead **contains information about both magnitude *and* direction**. ##### Distance traveled - **The term "distance traveled" is used to denote how long the path that an object took to get to some position is.** This value is equal to **the integral of the absolute value of velocity vs time or the integral of speed vs time**. - An object that takes a winding route to some position will have a greater *distance traveled* than an object that traveled there in a perfectly straight line. - Distance traveled **cannot be negative since it is a magnitude only**. - *This is NOT to be confused with distance which refers to the magnitude of displacement.* $\newline$ $\text{Distance traveled }=\large\color{lightcyan}|\,\Delta x\,|$ --- ## position $\color{LightCyan}x$ - In physics, the **position of an object refers to it's x and y coordinates on some arbitrarily created axis.** Negative numbers are commonly used to* denote "down" or "to the left" and positive numbers *are usually used to denote* "up" or "to the right." - The position of an object is denoted as $x$ or sometimes $(x,y)$ and so on depending on the number of dimensions it exists in. - Position **can be negative** because it inherently indicates both a direction and a magnitude/distance. $\newline$ $\text{Position from velocity and acceleration: }\;\;\Huge\boxed{\large\color{LightCyan}x_f=x_i\space+\space v_{\small i}t\space +\space\frac{1}{2}at^2\;}$ $\newline$ >[!quote] >##### *Position vs time* >- The **integral** of a position vs time graph isn't useful or important**. >- The **derivative** of a position vs time graph is **velocity vs time**. The slope of the tangent at some time $t$ is equal to the **instantaneous velocity of the object at that time**. >- The **x axis** usually represents the object's **starting position**. ></br> > >![[Motiondisplacementgraph.png|855]] --- ## Calculator <iframe src="https://app.calconic.com/api/embed/calculator/648a5a1828c7470029d3ebfb" 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="1000" scrolling="no" style="width: 100%; border: 0; outline: none;"></iframe>