2-Dimensional Motion: Difference between revisions
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<math>\vec{v} = \frac{\Delta \vec{r}}{\Delta t} = \left\langle \frac{\Delta x}{\Delta t},\frac{\Delta y}{\Delta t} \right\rangle</math>, so moving for <math>t</math> seconds with velocity <math>\vec{r}</math> can be represented as <math>\vec{v} \cdot t = \left\langle \frac{\Delta x}{\Delta t} \cdot t,\frac{\Delta y}{\Delta t} \cdot t \right\rangle</math> | <math>\vec{v} = \frac{\Delta \vec{r}}{\Delta t} = \left\langle \frac{\Delta x}{\Delta t},\frac{\Delta y}{\Delta t} \right\rangle</math>, so moving for <math>t</math> seconds with velocity <math>\vec{r}</math> can be represented as <math>\vec{v} \cdot t = \left\langle \frac{\Delta x}{\Delta t} \cdot t,\frac{\Delta y}{\Delta t} \cdot t \right\rangle</math> | ||
===Moving with constant acceleration=== | |||
In 2-dimensional space acceleration is <math>\vec{a} = \frac{\Delta \vec{v}}{\Delta t}= \left\langle \frac{\Delta v_x}{\Delta t},\frac{\Delta v_y}{\Delta t} \right\rangle</math>. | |||
A standard formula for position update is still applicable for vectors in 2-dimensional space: <math>\vec{r}(t)=\vec{r}_0+\vec{v}_0 t+\tfrac{1}{2}\vec{a}\,t^{2}=\langle x_0+v_{0x}t+\tfrac{1}{2}a_x t^{2},\ y_0+v_{0y}t+\tfrac{1}{2}a_y t^{2}\rangle</math> | |||
==Computational models== | |||
Revision as of 18:50, 2 December 2025
Kseniia Suleimanova Fall 2025
The Main Idea
When objects move in 2-dimensional space, their motion can be described in [math]\displaystyle{ \hat{x},\ \hat{y} }[/math] coordinates. The motion for each of those axes can be viewed independently. Another approach is using vectors (i.e. coordinate (5, 3) can be seen as vector [math]\displaystyle{ \langle 5,\ 3 \rangle }[/math]).
Displacement and distance
Imagine we have 2 points and origin O: A = [math]\displaystyle{ \langle a,\ b \rangle }[/math], B = [math]\displaystyle{ \langle c,\ d \rangle }[/math]
A point particle moves from origin to point A and then to point B. The displacement can be viewed as adding those vectors: [math]\displaystyle{ \langle a,\ b \rangle + \langle c,\ d \rangle = \langle a + c,\ b + d \rangle }[/math].
The distance is the sum of magnitudes of those vectors: [math]\displaystyle{ \sqrt{a^{2} + b^{2}} + \sqrt{c^{2} + d^{2}} }[/math]
Moving with a constant velocity
Velocity is the derivative of the position vector. In 2-dimensional space velocity looks the following: [math]\displaystyle{ \vec{v} = \frac{\Delta \vec{r}}{\Delta t} = \left\langle \frac{\Delta x}{\Delta t},\frac{\Delta y}{\Delta t} \right\rangle }[/math], so moving for [math]\displaystyle{ t }[/math] seconds with velocity [math]\displaystyle{ \vec{r} }[/math] can be represented as [math]\displaystyle{ \vec{v} \cdot t = \left\langle \frac{\Delta x}{\Delta t} \cdot t,\frac{\Delta y}{\Delta t} \cdot t \right\rangle }[/math]
Moving with constant acceleration
In 2-dimensional space acceleration is [math]\displaystyle{ \vec{a} = \frac{\Delta \vec{v}}{\Delta t}= \left\langle \frac{\Delta v_x}{\Delta t},\frac{\Delta v_y}{\Delta t} \right\rangle }[/math].
A standard formula for position update is still applicable for vectors in 2-dimensional space: [math]\displaystyle{ \vec{r}(t)=\vec{r}_0+\vec{v}_0 t+\tfrac{1}{2}\vec{a}\,t^{2}=\langle x_0+v_{0x}t+\tfrac{1}{2}a_x t^{2},\ y_0+v_{0y}t+\tfrac{1}{2}a_y t^{2}\rangle }[/math]
Computational models
Examples
History
See Also
Videos
https://youtu.be/V1I-vrXGl3A?si=G30q3yECGvbwonP7