When do you have motion in two dimensions?

If you neglect the fact that soccer players jump once in a while, you could simplify the description of their motion during a mach by limiting the analysis to only two dimensions. Another example that allows you to identified the motion as a two dimensional motion is the walking of a person in a flat city (without going to other floors of buildings). Even when the previous simplification seem to be drastic, still those motions will be hard to put in an exact mathematical form. The following descriptions can be applied to motions in two dimensions; while, in addition, the mathematical formulations can be easily extended to more than two dimensions.

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General Formulations for Motion in two Dimensions

In two dimensions the vector nature of the displacement, velocity, and acceleration become crucial. In this case, the displacement vector can be written as where and are the unit vectors along the horizontal and vertical axis. See Vectors. These two units vectors are fixed to the axis and do not depend on the time. However, the coordinate values x and y may be time dependent and, consequently, the displacement vector may be time depending.

The definition of average velocity is now

where is the final position of the moving object and is the initial position. As in the case of motion in one dimension, is the change in time or time taken. The instantaneous velocity is obtained when → 0,

It is assumed that both limits exist in order for the vector velocity to be well defined.

The average acceleration of the object also is defined similarly to the definition used for motion in one dimension,

where is the velocity at the time instant tf (final velocity) and the velocity is the velocity at the initial time ti (initial velocity). Just as in the case of the velocity, the instantaneous acceleration is obtained in the limit when the time taken is very small (→ 0),

The previous set of definitions can be used for describing motion in two dimensions.

 

 

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by Luis F. Sez, Ph. D.    Comments and Suggestions: LSaez@dallaswinwin.com