kinematics - What do negative times and negative displacements represent in parametric projectile equations?

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?

Answer

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".