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I have been lerning about acceleration and free fall recently, and we were told the formula for distance is $s=s_0+v_0t+\frac {1}{2}at^2$ (for movement along $x$-axis) but when we change the axis (to movement along $y$-axis) and describe the free fall equation it will now be like this $s=s_0+v_0t-\frac {1}{2}at^2$. My question is, if I have any formula for something (now generally, not talking about the example above, for any movement or anything) along the $x$-axis and I want to describe the movement (or anything else) along the $y$-axis, how do i know, when to change the + for - (or the quantify in this case $+g$ for $-g$)? Please explain for this case and also generally so that i can use the knowledge in the future when talking about different topics/problems.

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The equation $x = x_0 + v_0t + 1/2 at^2$ is the general kinematic equation that describes motion under constant acceleration. For motion in two or three dimensions, it is customary to use "x" for the horizontal direction, "y" for the vertical direction, and "z" for the "depth" direction. Thus, if an object is falling straight down, with no horizontal or z-directed motion associated with it, the equation to describe this motion would normally be $y = y_0 + v_0t + 1/2at^2$, where "a" is the acceleration due to gravity. Note that if "x" and "y" motion is occurring simultaneously (e.g., projectile motion), and equations of motion are being used to represent vector components, it is customary to use subscripts on the variables "v" and "a" to indicate which component variables are being described.

Regarding the direction of the motion, you are free to choose that direction at the start of the problem, and as long as you stay consistent in the sign designation, everything will work as it should. For most problems, up will be designated as positive and down will be designated as negative. Under this directional assignment, the acceleration due to gravity is negative, because it is in the negative direction. On the other hand, some problems may work better if down is defined as the positive direction. If that is the case, then the acceleration due to gravity is positive, and any motion in the upwards direction is negative.

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  • $\begingroup$ But then shouldn't the velocity also be negative? $\endgroup$ – Lauren Sin Dec 27 '18 at 23:47
  • $\begingroup$ Displacement, velocity, and acceleration are ALL vector quantities. The defined positive direction applies to all vector quantities, so if you defined up as positive, anything that is in the downward direction is negative. $\endgroup$ – David White Dec 28 '18 at 1:38
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Very generally, everything is a choice. We choose to favor one direction, or one sign, over the other. Often times, though, we will choose a convention that makes the most sense.

For example, it could make sense for the sign to be negative in the $y$-axis because it reminds us that the height of the object is decreasing with time. Conventionally, the positive direction of the $y$ axis is "up" and the negative direction is "down".

These choices eventually become the convention, and everyone agrees that that choice is the choice to use. It doesn't really matter which choice it is, so long that everyone is using it, consistiently. This is actually kind of weird because you already know a lot of conventions, but you've never even thought about it. Everyone knows that $x$ and $y$ are used as variables, while $a$, $b$, and $c$ are used as constants, and it just looks strange if you don't do it like that. Conventions are sometimes silent agreements that are hardly touched upon or "taught"; those are the ones that will almost never cause confusion.

Other times, the choice is in the hands of the teacher or author, for their own reasons. In these cases, you have to go and ask them what convention they are using (or read the "about this book" section). In your case, ask the teacher: "Hey, is this negative sign always going to be the case?" and make sure that you understand how they are choosing to express their equations or signs so that there are no misunderstandings.

Similarly, if you are talking about or teaching a topic and you're choosing to use some unnatural conventions, be sure to let others know, so you that they can understand you.

Ultimately, the universe doesn't care. The answers should be the same, regardless of which ones we choose.

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