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I am parametrizing equations of motion in the form: $$x(t) = x_0+v_{0,x}t\\y(t) = y_0+v_{0,y}t+\frac{1}{2}at^2$$ The parametrized equation with respect to time: $$y(x) = y_0+v_{0,y}\cdot \frac{x - x_0}{v_{0,x}}+\frac{1}{2}a\cdot \left(\frac{x - x_0}{v_{0,x}}\right)^2.$$ The interpretation of the last equation is trajectory.

What I struggle with is the interpretation for similar equations expressed for velocities and acceleration. Is there any interpretation of $v_y(v_x)$ and $a_y(a_x)$?

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The x component of velocity and acceleration are constants; you don't use them as function variables. Even if you treat them as variables, the y component and the x component of both velocity and acceleration are independent of each other; the change in the x component makes no difference to the y component. These are not valid functions

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  • $\begingroup$ Therefore they have no physical meaning? This is what you imply here? $\endgroup$
    – Radek D
    Commented Nov 21, 2023 at 15:33
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    $\begingroup$ yes. They do not have physical meanings like y(x) $\endgroup$
    – Y Z
    Commented Nov 22, 2023 at 8:12

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