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We expect a truly general relation, such as Eqs. 2-26 and 2-28, to be valid regardless of the choice of coordinate system. By demanding that general equation be dimensionally consistent we ensure that the equations are valid regardless of the choice of units. It there any need then for units or coordinate systems?

We expect a truly general relation, such as Eqs. 2-26 and 2-28, to be valid regardless of the choice of coordinate system. By demanding that general equation be dimensionally consistent we ensure that the equations are valid regardless of the choice of units. It there any need then for units or coordinate systems?

Grade:11

1 Answers

Kevin Nash
askIITians Faculty 332 Points
9 years ago
Below, we show the kinematic equations describing the motion of a particle along the ‘x‘ direction, moving with initial velocity v0x , during time interval t , and with constant acceleration ax using the Cartesian coordinate system.
v_{x} = v_{0x} + a_{x}t
x = x_{0} + v_{0x}t + \frac{1}{2} a_{x}t^{2}
It is important to note that the equations defining the general relation (given above) are dependent on the dimensions of the coordinate system. As long as the reference frame of the observer is same, the perception of events would not alter. A different coordinate system would describe the same motion with different dimensions and henceforth the equation would modulate accordingly.
For example, an observer locating the position of an object using the polar coordinate system will eventually do so in dimensions such as the angle\theta with the horizontal axis and position vector r . Whereas the observer locating the position of the same object using the Cartesian coordinate system will need two dimension such as x and y respectively.
Therefore the equation defining the location of the object modulates with the coordinate system, however both withdraws the same conclusion.
Be defining the units, the observer accounts for the dimensionally consistence of the general equations. Therefore the units play an important role for verifying the relation.
For emxaple, in equationv_{x} = v_{0x} + a_{x}t , one can see that the units on the left side of the equation match that with the units on the right side, which is m / s (see below):

\left [ m/s\right ] = \left [ m/s\right ]+ \left [ m/s^{2}\right ] \left \left [s] \right
\left [ m/s\right ] = \left [ m/s\right ]+ \left [ m/ s \right ]
\left [ m/s\right ] = \left [ m/s\right ]

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