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There are negative times and negative $x$ and $y$ displacement in parametric equations; what do they represent? And is it possible to tell the object's velocity, direction and acceleration from just a parametric equation?

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I am going to assume that:

We have three variables:

  • $x(t)$, the horizontal displacement from the origin, as a function of time
  • $y(t)$, the vertical displacement from the origin, as a function of time
  • $t$, the time passed from the start of the motion

I'm assuming you mean displacement instead of distance, because a negative distance has no meaning. A displacement is similar to distance, except displacement describes direction as well. A negative displacement is the same as a positive displacement, except in the opposite direction. For example, say the if a horizontal displacement of +4 corresponds to "4 to the right". This means a horizontal displacement of -4 corresponds to "4 to the left".

Now, you need to ask yourself the question: for what range of values of $t$ is $x(t)$ and $y(t)$ going to be valid? Equations are usually modelled after a physical motion for a finite range of $t$. Sometimes, equations are defined only for $t>0$, so the values for $x(t)$ and $y(t)$ for $t<0$ have no meaning. An example of this is when $x(t)$ and $y(t)$ describes the position of a cannonball as it is fired through the air, but the equation falls to pieces at any time before the cannonball is launch.

However, you will get cases where $x(t)$ and $y(t)$ is defined for $t<0$. In these cases, $t<0$ refers to time before the "start of the motion". Even though $t=0$ marks the "start of the motion", that doesn't necessarily infer that there was no motion before that point in time, provided $x(t)$ and $y(t)$ are sufficiently defined.

Therefore, if you look at $x(t)$ and $y(t)$ for $t<0$, you will be looking at a previous position of the object before it has reached the point in time that is selected to be the "start of the motion".

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  • $\begingroup$ That's right. Just an attempt to sum it up briefly: mathematics if our tool to solve the problem. Math justs sees "a parabola", but mathemathics doesn't care if it make sense. It's you the one who must be careful and use the math within the scope of the physical truth. $\endgroup$ – FGSUZ Jun 29 '18 at 11:56

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