Question

if the question ask about free fall, how can we assume that the gravitational acceleration is positive or negative? for example , if i take g positive towards the earth, will the displacement will also be positive?

Answer 1

the most common convention is to say that +ve displacement is up or away from the earth, and to include gravity as -9.8m/s/s

but if you wish your coordinate system to say that +ve displacement is down or toward the earth, then be sure to include gravity as +9.8m/s/s

to test this, if you are using one of the equations of motion such as \(x = u_o t + \frac{1}{2} a t^2\), you will see that specifying a = g means that +ve displacement is toward the earth. it has to be as the thing in question will fall toward earth if under only the influence of gravity. but if you specify a = -g, +ve displacement has to be away from the earth.

and, if you're building it from calculus you will see the same, for example: \(\ddot x = -g \implies \dot x = u_o - gt \implies \Delta x = u_o t - \frac{1}{2} g t^2\)

Answer 2

First of all, gravitational acceleration is a vector quantity which always acts towards the earth. When you draw a diagram o a freely falling object, g must always be directed towards the earth (downwards), but then, its your wish whether you take the downward direction to be positive or negative, which will not affect the ultimate result of any numerical problem. The standard accepted convention is that, the upward direction should be taken positive and the downward direction should be taken negative.