Free fall — g = 9.8 m - s², sign conventions: Foundations

This page assumes you have seen nothing. We build every letter, symbol and picture the parent note Free-fall page uses, in the order that lets each one lean on the one before it.

0. What a "position axis" even is

Before we can say where a ball is, we need a ruler pinned to the world. Pick a point to call zero, draw a straight line through it, and mark off equal steps ( $1, 2, 3, \dots$ metres) in one direction and ( $- 1, - 2, - 3, \dots$ ) in the other.

Figure — Free fall — g = 9.8 m - s², sign conventions

The direction you chose as positive is your sign convention. Everything below depends on it.

1. Direction as a sign: $+$ and $-$

On our ruler, the arrow direction (say, upward) is $+$ and the opposite (downward) is $-$ . A quantity with both a size and a direction-along-the-line is the simplest kind of vector, and on one straight line a vector is just a number with a sign.

2. Time: the symbol $t$

3. Velocity: the symbol $v$ and the ratio behind it

Watch the ball move a small distance $Δ y$ (the symbol $Δ$ , a Greek "D", means "change in" — the end value minus the start value) during a small time $Δ t$ .

As we let $Δ t$ shrink toward an instant, this ratio becomes the instantaneous velocity, written $\frac{d y}{d t}$ (see §5).

4. Acceleration: the symbol $a$ and why $g$ is special

Velocity itself can change. Acceleration measures how fast the velocity changes.

5. The derivative $\frac{d v}{d t}$ — why calculus appears

The parent writes $a = \frac{d v}{d t}$ . That's the ratio of §3–4 taken to the limit of an instant rather than a chunk of time.

6. The force–mass link: $F = ma$ and why mass cancels

Gravity pulls with a force $F = m g$ . Put that into $F = ma$ : $ma = m g ⟹ a = g .$ The $m$ cancels — appears on both sides, divides out. That is why a hammer and a feather fall identically in vacuum: more mass feels more pull, but also resists more, and the two exactly balance.

y

represent, and why not

x

A number giving position along a ruler; we use

y

because free fall is vertical.

What does

Δ

(Greek D) in front of a quantity mean?

"Change in" — the final value minus the initial value.

On a "up is positive" axis, what does a negative velocity mean physically?

The object is moving downward (sign = direction, not slowing).

What is the difference between

g

and

a

g

is a fixed positive size (

9.8

);

a

is the acceleration whose sign (

+ g

- g

) depends on your convention.

Why is velocity defined as a ratio

Δ y /Δ t

It answers "how much position per unit time" — capturing distance and duration at once.

What does the derivative

d v / d t

give that

Δ v /Δ t

does not?

The instantaneous rate of change at one exact moment, not an average over a chunk.

F = ma

with gravity

F = m g

, why does mass cancel?

Both sides carry an

m

; dividing it out leaves

a = g

, so acceleration is independent of mass.

What does the subscript

0

v_{0}

y_{0}

mean?

The value at the start, when

t = 0

(initial velocity / initial position).

Connections

Free fall — g = 9.8 m - s², sign conventions — the topic these foundations build toward
Vectors and sign conventions — where signs-as-directions comes from
Equations of Motion (constant acceleration) — the integration that produces the kit
Newton's Second Law — source of $a = g$
Projectile Motion — reuses every symbol here for the vertical part

Foundations — Free fall — g = 9.8 m - s², sign conventions