# Free fall and projectile motion

I'm wondering if something is falling say from a roof, would the distance it falls be the final $y$ position? Also would all the $y$ components (velocity, displacement) be negative?

You are free to pick your frame of reference. You can point $y$ up or down, or even sideways. You can put the origin at the top of the roof, at ground level, or at the center of the earth.

My recommendation - in problems like you are describing, ALWAYS draw a diagram that shows what conventions you use - after that, you essentially answer your own question.

For example, if you put $y=0$ at the top of the roof, and the final position is $y=-50$, you can say the object fell 50 m. But if you had put the roof at $y=60m$, and the object had fallen onto the ground at $y=10m$ (you had put the reference level in the basement of the building, two floors below "ground") the distance would still be 50m, not 10 m (even though that was the final position).

Key here is that coordinate systems are arbitrary, but distance is a relative measure - so it involves subtracting two coordinates. That gets rid of the problem with the choice of origin.

• Ok. so to follow up with this, to find the velocity in the y direction of something falling from a building as a projectile,would one use cosine instead of the regular sine when given the height of the building and the ultimate goal is to find where the object that fell from the top of the building landed? Sep 21, 2014 at 21:30
• Not sure where angles are suddenly coming in... Look at horizontal and vertical velocities separately. If you roll off a slanted roof you have a certain horizontal velocity (which is usually assumed to remain constant after leaving the roof) and a vertical component (which will change as the object accelerates). Sine or cosine depends on your definition of the angle... See comment about drawing a diagram. Sep 21, 2014 at 23:27
• +1 for the diagram recommendation; that always helps. Great answer. Sep 26, 2014 at 0:16

It depends on what my physics teacher might call your "direction of reference". Sure, you can make "down" positive. In this case, there shouldn't be any preferred direction along the $y$ axis - down is just a valid a positive direction as up - that is, as long as you adjust the other variables to match. For example, if the acceleration is $g$, you would say that $a = 9.8$ $m/s^2$, instead of $-9.8$ $m/s^2$. Also, for an object falling down, $v$ (so $v_0$ and $v_f$, and all the $v$s in between) would be positive, as long as the object continues falling down. Displacement, velocity, and acceleration - as long as they are all "downwards". Overall, though, designating "down" as negative or positive is simply a matter or preference. That said, there are scenarios where choosing a different sign convention might make things easier - as in this case, if the object continues moving only in one direction.

A recent edit to this question by @QMechanic put me thinking on another path that might help me explain this better. In many physics problems and theories, you can use different coordinates. For example, in general relativity (and Riemannian geometry in general), you can use all different types of coordinate systems. For instance, in flat space, polar and Cartesian coordinates are equally valid. However, if you want to look at metrics involving distances from a single point, polar (2 spatial dimensions) or spherical (3 spatial dimensions) might be better than Cartesian coordinates. It's all a matter of preference, but some ways are easier than others.