>[!summary] Motion in 1D is used to describe motion as a function as time. > We assume displacement to the change in distance over time, which is the integral of velocity > We assume velocity as the rate of change of displacement or the integral of area. > We assume acceleration as the rate of change of velocity. > **General equation:** $\Delta x = v_{avg} t$ $\Delta v = a_{avg} t$ > **Projectile motion:** Motion in two dimension usually created by some force with an initial force upwards. > General equation to break up into two dimension: >$\begin{array}{c} X - direction \\ \Delta x = v_{avg} t \\ \Delta v = a_{avg} t \\ \\ Y - direction \\ \Delta Y = v_{yavg} t \\ \Delta v_y = a_{y avg} t \\ \end{array}$ >[!info]+ Read Time **⏱ 3 mins** # Motion in 1D Kinematics is a description of motion in 1D such as displacement, velocity, and acceleration. Displacement is how a object will move as a function as time. Displacement can also be found by integrating velocity, seeing how the area of velocity changes as a function of time. Velocity is the rate as displacement at a function of time, or the speed at which this happens. Velocity can also be found by seeing how the area of acceleration changes as a function of time. Acceleration is the rate of which velocity changes as a function of time. >[!info] In general this is true mathematically: Displacement = x(t) or $\int y(t)$ Velocity = v(t) = $\frac{dx}{dt}$ or $\int a(t)$ Acceleration = a(t) = $\frac{dv}{dt}$ ![[kin_1.png]] [^1] >[!note] Explanation Example of motion in 1D In motion in 1D we can describe motion as an effect due to velocity and acceleration. >[!info] Important The object **speeds up** when velocity and acceleration are the in the **same direction** The object **slows down** when the velocity and acceleration are in the **opposite direction**. We often describe 1D as graphs like in the figure, which can be useful to view how velocity acceleration and displacement change with time. ![[kin_2.png]] ![[kin_3.png| 400]] [^1] >[!note] Explanation Figure to graph displacement, velocity and acceleration That figure graphs as a function of displacement velocity and acceleration # Deriving General Expressions >[!warning] Assumptions We will assume that in a displacement, velocity and accelerations case, the total displacement, is over the total distance is the integral of velocity over total time. (Same for acceleration) In the case of finding displacement: ![[kin_4.png]] [^1] >[!note] Explanation Total displacement is the area underneath the velocity. The total displacement can be found as this: $\begin{array}{c} \Delta x = \int_t ^ T v(t) dt \\ \Delta x = v_{avg} t \end{array}$ Same idea can be done for velocity: ![[kin_5.png]] [^1] >[!note] Explanation Total velocity is the area underneath the acceleration. $\begin{array}{c} \Delta v = \int_t ^ T a(t)dt \\ \Delta v = a_{avg} t \end{array}$ # Projectile Motion In projectile an object would have been giving a velocity upwards at some angle in normally two dimensions (x,y) ![[kin_6.png]] [^1] >[!note] Explanation Example of projectile motion over a distance. In general, we would assume separate equations for two dimensions: $\begin{array}{c} X - direction \\ \Delta x = v_{avg} t \\ \Delta v = a_{avg} t \\ \\ Y - direction \\ \Delta Y = v_{yavg} t \\ \Delta v_y = a_{y avg} t \\ \end{array}$ [^1]: Taken from R. Epp Lecture notes. --- 📂 Want to see more structured notes like these? Help grow the project by [starring Math & Matter on GitHub](https://github.com/rajeevphysics/Obsidan-MathMatter). ---