#mainpage --- ## What is acceleration $\large\color{LightCyan}a$? ##### Definition - Acceleration is **the rate at which velocity changes with time** *(usually measured in $m/s^2$)*. It denotes **both the magnitude and direction of change**, *negative for decreasing velocity, positive for increasing velocity.* ##### Properties - Acceleration is the slope or derivative **of an objects *velocity*.** - The [[Integration|indefinite integral]] of a function that **describes an object's acceleration vs time** will give you a function that describes the **object's velocity over time.** - Taking the [[Integration|definite integral]] of an object's acceleration vs time function will give you the **net change in that object's velocity over the specified bounds.** - Acceleration is a **vector quantity** and therefore denotes both a **magnitude and a direction**. This means that an object's *acceleration* can be **either negative or positive**. - A **negative acceleration** usually means the object is accelerating towards the *left or down* *(in a traditional 2d system)*. - A **positive acceleration** indicates the object is accelerating towards the *right or up* (in a traditional 2d coordinate system). - **The result of this is somewhat unintuitive:** an object traveling right will decelerate only when its **acceleration is negative** and an object traveling **left will only decelerate if it's acceleration is positive.** >[!danger] >##### *Velocity and acceleration* >- **Just because an object's velocity is zero does not mean acceleration is zero.** >- For example, *the moment* when a thrown ball reaches its max height and switches directions you might be tempted to say the acceleration is zero. However, this isn't true, the acceleration **would be negative, in fact it has been negative for the entire throw** *(assuming up = positive and down = negative).*** $\newline$ $\text{Acceleration: }\;\Huge\boxed{\normalsize\space\color{LightCyan}a=\frac{\Delta\space v}{\Delta\space t}\;}\;\;\;\normalsize\text{Average acceleration: }\;\Huge\boxed{\normalsize\space \color{LightCyan}a_{\text{avg}}=\left(\frac{v_f-v_i}{\Delta\space t}\right)\;}$ $\newline$ $\text{Derived acceleration equations:}\;\;\Huge\boxed{\normalsize\;\color{lightcyan}a=\frac{x_{\small f}-x_{\small i}-v_{\small i}\cdot\Delta t}{0.5\cdot \Delta t^2\;}}\;\;\;\boxed{\normalsize\;\color{lightcyan}a=\frac{v_{\small f}^2-v_{\small i}^2}{2\cdot\Delta x}}$ $\newline$ >[!check] >##### *Acceleration sign and slope cheat sheet* >| **Sign of acceleration** | **Slope of acceleration** | **Resulting motion** | >|-|-|-| >| Positive | Positive | **Accelerating to the right.** | >| Positive | Negative | **Decelerating to the right** | >| Negative | Negative | **Accelerating to the left** | >| Negative | Positive | **Decelerating to the left** | >[!quote] >##### *Acceleration vs time graph tips* >- **The area _(or integral)_ under an object's velocity vs time graph is equal to its net change in velocity.** > - The slope *or derivative* or an object' velocity vs time graph is not a quantity that's typically used by physicists. Despite this it can still give you information about how an object's velocity/position is changing *(see "check" box above)*. >- If an object's acceleration is **equal to zero** that indicates the object has **constant velocity**. Non-zero constant acceleration indicates **that the object's velocity is either linearly increasing or decreasing.** > - Linear, non constant acceleration indicates **an exponential change in velocity**. >- If an objects acceleration switches signs the object has gone **from either speeding up, to slowing down or slowing down, to speeding up.** > ></br> > >##### *Motion graphs compared* >![[Accelerationvstimegraph.png|850]] --- ## Acceleration and force - To see more about force please refer to [[Forces|this page on force]]. - Since acceleration is a component of **newton's second law** *(the force equation)* we can derive a few equations for the acceleration of an object based on the mass of an object and the force that is being exerted on it. - Acceleration is also involved in **circular/rotational motion equations** for more information on this please refer to: [[Rotational and Circular motion|this page]]. $\text{Acceleration from Newton's second law: }\;\;\Huge\boxed{\normalsize\color{lightcyan}\;a=\frac{F}{m}\;}$ --- ## Calculator <iframe src="https://app.calconic.com/api/embed/calculator/648bf21528c7470029d3f09c" 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>