2-Dimensional Motion
PAGE CLAIMED BY SUHAILA RASHID (FALL '24)
The motion of an object can occur in one dimension, two dimensions and three dimensions. One dimensional motion occurs along one axis such as x. Two dimensional motion occurs when an object moves along two axes, such as x and y. Three dimensional motion occurs in the three axes of motion: x, y and z, which provides the most accurate representation of the reality.
The Main Idea
Two Dimensional Motion is a model to extrapolate the properties of an object moving along two axes, usually x axis and y axis. The properties of an object include it's position, velocity and acceleration. Two dimensional motion has two vector components derived from each axis of motion.The x component of motion is independent form the y component of motion with time relating the two components.
A Mathematical Model
- [math]\displaystyle{ \begin{align} v & = at+v_0 \quad [1]\\ \end{align} }[/math]
- [math]\displaystyle{ \begin{align} d & = d_0 + v_0 t + \frac{{a}t^2}{2} \quad [2]\\ \end{align} }[/math]
- [math]\displaystyle{ \begin{align} d & = d_0 + \left( \frac{v+v_0}{2} \right )t \quad [3]\\ v^2 & = v_0^2 + 2a\left( d - d_0 \right) \quad [4]\\ d & = d_0 + vt - \frac{{a}t^2}{2} \quad [5]\\ \end{align} }[/math]
Examples
Simple
Dr.Greco stands on a 50ft high cliff in his penguin suit and throws a ball his friend who is standing x number of feet from the cliff.How far from the bottom of the cliff is his friend if it takes 10 seconds for the ball to reach his friend and the initial velocity is 20m/s?
You have to know that the x component of motion is independent form the y component of motion.
- [math]\displaystyle{ \begin{align} d & = d_0 + vt \quad \end{align} }[/math]
Middling
Dr.Greco stands on a 50ft high cliff in his penguin suit and throws a ball his friend who is standing 50 ft away from the cliff. What is the y component of velocity if it takes 10 seconds for the ball to reach his friend and the initial velocity is zero.
- [math]\displaystyle{ \begin{align} d & = d_0 + vt - \frac{{a}t^2}{2} \end{align} }[/math]
Use the formula above with the knowledge that velocity is zero and the acceleration is equal to gravity to determine that the y component of the velocity has to be equal to get the y component of final velocity.
- [math]\displaystyle{ \begin{align} d & = \frac{{a}t^2}{2} \quad \end{align} }[/math]
Difficult
Dr.Greco stands on a 50ft high cliff in his penguin suit and throws a ball, with a velocity of 20 m/s, to his friend who is standing 50 ft away from the cliff. What is the x and y component of velocity?
Use this equation to find the time because time is the component that relates the motion along the two axes.
- [math]\displaystyle{ \begin{align} d & = d_0 + vt \quad \end{align} }[/math]
Then, use this to find the x and y component of final velocity.
- [math]\displaystyle{ \begin{align} d & = d_0 + vt - \frac{{a}t^2}{2} \quad \end{align} }[/math]
History
Motion of an objects has been studied since the time of Aristotle. However, it was not until Galileo's experimentation with inclined planes did we really discover the concept of 2 D motion.1
See also
Further reading
http://www.physicsclassroom.com/class/vectors http://physics.bu.edu/~duffy/py105/Motion2D.html
External links
https://www.khanacademy.org/science/physics/two-dimensional-motion
References
1.http://galileo.rice.edu/lib/student_work/experiment95/paraintr.html 2.https://en.wikipedia.org/wiki/Equations_of_motion