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In kinematics, the important variables are the velocity v, the acceleration a, and the object's position x. These variables are usually presented as functions of time: x(t), v(t) and a(t).

The acceleration can either be constant, or vary with time, i.e. a(t), or vary with position, a(x) or even vary with velocity, i.e. a(v).

An object that moves occupies different spatial positions x at different instants of time t. If the acceleration a(t) varies with time, the object will have a different accelerations at every different spatial position, we can interpret the acceleration also as a(x). But an acceleration that depends on time and one that depends on space are different things. Parametrically, we can always express the acceleration in terms of any other independent variable (t, v, x). But how can we distinguish between the different cases where the acceleration has a time or space dependence that is explicit or implicit if parametrization can always convert the function from one to the other?

For example, say a(t) = 3t. We solve for x(t) and express t(x). The replace t in the equation for a(t) and get a(x). That is different from starting with an acceleration a(x)...

Thanks.

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# I Parametrization and kinematics

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