Which cases $g$-force is negative $(g=-9.81m/s^2)$? 
A stone was dropped into an empty well.  It takes 10 sec to hit the ground. What was the velocity of the ball?


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*So since I have to use the formula $v=gt$ I'm quite confused if g=$-9.81m/s^2$ or not? If g is negative, velocity will be negative too right?

*I also want to know if the stone was thrown vertically upward from the ground do we take g=$-9.81m/s^2$?

It just gets me confused when $g$ should be $-9.81m/s^2$.
 A: It totally depends on  how you set your coordinate axes but remember to apply the rules of your selected cordinate to all the physical quantities in action.
CASE 1 : if you chose your position as origin and downward direction as $(-)ve \;Y \; axis$ then everything in that direction will hold a $(-)ve$ sign before it's magnitude like velocity in the downward direction will be $-v$ and acceleration due to gravity in the same downward direction will be $-g$. Since you chose the downward direction as $(-)ve$, the upward direction ultimately becomes $(+)ve$ and hence if your velocity is in upward direction, it will be written as $+v$.
CASE 2 : And if you set the downward direction as $(+)ve$ then all physical quantities like velocity , acceleration ,  etc. in the downward direction will be considered $(+)ve$ and the quantities in the upward direction ultimately becomes $(-)ve$.
Since $g$ always points towards the center of the earth , it can be either $+g$ (if you take the downward direction as $(+)ve$  or $-g$ (if you take the downward direction as $(-)ve$.
Hope it helps ☺️.
A: $g\approx 9.81$ m/s$^2$ should never be negative.  In this context, $g$ is just shorthand.  If you define your coordinates such that up is the positive $y$-direction, then the vertical component of the acceleration is $a_y = - g$ and the equation to use is $v_y = - gt$.
If you would find it more useful to define down to be the positive $y$-direction, then the vertical component of the acceleration is $a_y = g$ and the equation to use is $v_y = gt$.
A: It depends on your choice or axis. $g$ is directed downward. If your axis is also directed downward, then $g$ is positive. Otherwise it's negative.
A: If you take it to be negative, then you're defining the upward direction to be positive and so all vectors pointing in the upward direction will be positive and all vectors pointing in the downward direction will be negative. You could have done it the other way round. You could have called the downward direction positive, then g would be positive and all vectors pointing in the upward direction will be negative.
Which direction you call positive and negative will have no effect on the physics. For example when you use $v = gt$, $v$ is the velocity of the ball dropped from rest after time $t$. So the vector $v$ will point downwards because the ball is moving downwards. Because g also points downwards, both vectors will have the same sign, positive or negative depending on which direction YOU choose to be positive. The sign only tells you the direction. The magnitude of $v$ will be unchanged.
When a stone is thrown vertically upwards, and suppose you want to find the maximum height. If you pick the upward direction as positive, (and hence g negative), then the height will come out to be positive because what you're really calculuating is the displacement vector which starts from the ground and points to the maximum height, so it points upwards. If you take g to be positive, the height will come out negative because you're taking the downward direction positive.
So to sum it up, g is neither positive nor negative. It's sign depends on your convention.
