<p align="right">Last Update: <font color="#4f81bd">December 12, 2024</font></p> ## BIG IDEAS - Only use these equations when the [[1-Lesson Plans/Archived/Acceleration]] is constant (not constant velocity). > [!NOTE] No subscripts > Please note that $u$ is used as initial velocity so that I do not need to include subscripts in the equations. ### Equation 1 (Velocity -Time Relationship) This equation is an algebraic rearrangement of the definition of acceleration. $v \ = \ u \ + at \tag{1}$ where $v$ is the final velocity, $u$ is the initial velocity, $a$ is acceleration, and $t$ is the time interval. > [!NOTE] No position > Equation 1 does not include position. #### Case 1 If the velocity is constant, then the acceleration is zero, and the equation is $v \ = \ u \tag{Case 1}$ #### Case 2 If the acceleration and velocity are in opposite directions then the object is slowing down. For example, the initial velocity is east and the acceleration is west: $v \ = \ u \ + (-a) \cdot t \tag{Case 2a}$ OR For example, the initial velocity is west and the acceleration is east: $v \ = \ (-u) \ + at \tag{Case 2b}$ ### Equation 2 (Position-Time Relationship) $s \ = \ ut \ + \ \frac{1}{2}at^2 \tag{2}$ where $s$ is displacement, $u$ is initial velocity, $\vec{a}$ is acceleration, and $t$ is the time interval. > [!NOTE] No final velocity > Equation 2 does not include final velocity ($v$). #### Case 1 If the velocity is constant, then the acceleration is zero, and the equation is $s \ = \ ut \tag{2}$ Which is a definition of average velocity. #### Example Problem <div class="sp-embed-player" data-id="cZlI1pnnp7S"><script src="https://go.screenpal.com/player/appearance/cZlI1pnnp7S"></script><iframe width="100%" height="480px" style="border:0;" scrolling="no" src="https://go.screenpal.com/player/cZlI1pnnp7S?width=100%&height=480pc&ff=1&title=0" allowfullscreen="true"></iframe></div> ### Equation 3 (Velocity-Position Relationship) $v^2 \ = \ u^2 \ + \ 2as \tag{3}$ where $v$ is the final velocity, $u$ is the initial velocity, $\vec{a}$ is the acceleration, and $s$ is the distance. > [!NOTE] No time > Equation 3 does not include time ($t$). ### Slide Decks <div style="position: relative; 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width: 100%; height: 0; padding-top: 56.2500%; padding-bottom: 0; box-shadow: 0 2px 8px 0 rgba(63,69,81,0.16); margin-top: 1.6em; margin-bottom: 0.9em; overflow: hidden; border-radius: 8px; will-change: transform;"> <iframe loading="lazy" style="position: absolute; width: 100%; height: 100%; top: 0; left: 0; border: none; padding: 0;margin: 0;" src="https://www.canva.com/design/DAGVq1rcMX4/0Bc3gz1174VndJGtLaiJfA/view?embed" allowfullscreen="allowfullscreen" allow="fullscreen"> </iframe> </div> <div style="position: relative; width: 100%; height: 0; padding-top: 56.2500%; padding-bottom: 0; box-shadow: 0 2px 8px 0 rgba(63,69,81,0.16); margin-top: 1.6em; margin-bottom: 0.9em; overflow: hidden; border-radius: 8px; will-change: transform;"> <iframe loading="lazy" style="position: absolute; width: 100%; height: 100%; top: 0; left: 0; border: none; padding: 0;margin: 0;" src="https://www.canva.com/design/DAGYqdW73eg/VdfuOeE7uDBU08gYv0lvqQ/view?embed" allowfullscreen="allowfullscreen" allow="fullscreen"> </iframe> </div> <div style="position: relative; 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