Some confusion on distance formula for physics projectile motion

Question

Based on self-study, I have 2 formulas which I see used for calculating distance traveled given time, velocity and acceleration. One is: $s(t)=s_0 + v_0t-1/2 g t^2$

and the other is:

$d=v_i t + 1/2 a t^2$

One of these equations has a "+" sign in front of the $1/2at^2 $ and the other has a "-" sign in front of $1/2gt^2$. Both calculate distance. Can you help me understand the difference in the 2 equations. Because of this confusion, it's unclear when to use one equation vs the other. It would seem I would get difference answers to the same distance question.

For example, a question states that a vehicle traveling at 32mph begins to decelerate at a rate of $3 ft/s^2$. How far will it travel during the 15.6 seconds it takes to come to a complete stop?

I use the $d=v_it + 1/2at^2$ being careful to correct for mismatched units and to use a $-3$ for the $a$ value. I get the correct answer of $367$ ft.

If I had used the $s(t)=s_0 + v_0t-1/2 g t^2$ equation I would have gotten a different answer. I clearly don't understand these 2 distance equations. Can you assist in clearing up this confusion?

Answer 1

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Both equations are the same. What you should see is that they are just differing in the interpretation of you one-dimensional movement.

The first equation is

$$s(t)=s_0 + v_0t-1/2 g t^2$$

This equation is saying that the particle is moving from the initial point $s_0$ to the point $s(t)$ after a time $t$ has passed. The minus sign is saying about the accelerated nature of the motion. If there is a minus then the particle is getting less positive velocity and is getting slower and slower as time goes by. Then what you're getting is that the gravity is pushing the particle as a normal acceleration $a$ but in the opposite direction of the motion.

The other equation is just the above when $g \to -a$ where you get, passing from $g$ to a more general acceleration $a$,

$$s(t) - s_0 := d(t) = v_0 t + 1/2 a t^2$$

Here $d$ stands for 'displacement'. It is the distance traveled by the particle.

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Note your example, when you set the minus sign to represent decelerating movement you get the correct answer. This is because mathematically the minus sign is representing the physical meaning of decelerating. The plus sign, in the other case, stands for accelerated movement. So if you have a car that is accelerating (the driver is accelerating the car pressing down the pedal) you'll have a positive acceleration so you should use the formula with a plus.

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So will I get the same results?

Only if you use correctly the signs. These equations are the same, the first one only uses gravity as acceleration. The second one is more general, if you are accelerated use $+a$ but if you are decelerated use $-a$ in the second equation. The first equation is just the second one opened and with a minus gravity on it.