Projectile motion — horizontal - vertical independence, full derivation: Foundations

This page is a toolbox unpacking. The parent note (Projectile motion — horizontal - vertical independence, full derivation) throws around symbols like $F$ , $u cos θ$ , $a_{y} = - g$ , and $\int v d t$ as if you already know them. Here we earn each one, in build-order, with a picture attached.

1. Position, the origin, and the axes ( $x$ , $y$ , $O$ )

Look at the figure: $O$ is the bottom-left corner. A ball sitting at the dot is described by two numbers — walk right by $x$ , then up by $y$ . Those two numbers are its position.

Figure — Projectile motion — horizontal - vertical independence, full derivation

2. Time $t$ — the shared clock

We write $x (t)$ and $y (t)$ — read " $x$ at time $t$ " — to stress that position is a function of time: give me a $t$ , I give you back a position.

3. Velocity — an arrow with a length and a direction

4. The launch angle $θ$

Look at the next figure: the amber arrow is the launch velocity of length $u$ , tilted up by $θ$ from the white ground line.

5. Splitting the arrow: $sin θ$ , $cos θ$ , and components

This is the single most important tool the parent assumes. We build it slowly.

In the same figure the cyan dashed lines show these two sides. The sideways shadow is $u_{x}$ ; the vertical shadow is $u_{y}$ .

Every case of the split

6. Acceleration $a$ and gravity $g$

The two directions split perfectly here: $a_{x} = 0 (no sideways push), a_{y} = - g (gravity, downward) .$

7. Force $F$ and Newton's law, component by component

8. The slicing tools: $\frac{d v}{d t}$ and $\int v d t$

The parent writes $a = \frac{d v}{d t}$ and $x = \int v d t$ . Here is what those squiggles mean, from zero.

If constant acceleration feels more comfortable as ready-made formulas (no calculus), the algebraic route $v = u + a t$ , $s = u t + \frac{1}{2} a t^{2}$ lives in 1-D Kinematics — Equations of Motion — it is the same result integration produces.

9. The parabola and $tan θ$

Prerequisite map

Read it top-to-bottom: the raw ideas on the left/top feed the parametric equations $x (t), y (t)$ , and those in turn give every result the parent boxes.

Equipment checklist

Cover the right side and test yourself — you are ready for the full derivation when every one is instant.

What do

x

and

y

measure, and from where?

Sideways distance and height, both from the origin

O

What single thing do the horizontal and vertical motions share?

The clock — the same time

t

What does the arrow-hat

mean?

The quantity is a vector: it has direction, not just size.

What is

θ

measured from?

The horizontal ground line.

Write

u_{x}

and

u_{y}

in terms of

u

and

θ

u_{x} = u cos θ

u_{y} = u sin θ

Why cosine for the horizontal part?

Cosine is adjacent-over-hypotenuse; the horizontal side is adjacent to

θ

What is

a_{x}

and why?

0

, because there is no horizontal force.

Why is

a_{y} = - g

(why the minus)?

Up is positive; gravity points down, so its acceleration is negative on that axis.

Does mass appear in

a_{y}

No — mass cancels in

- m g = m a_{y}

, so all projectiles fall alike.

What does

a = \frac{d v}{d t}

say in words?

Acceleration is the rate at which velocity changes per unit time.

What does

\int v d t

do?

Adds up all the little

v d t

steps to give total displacement (velocity → position).

Why is the path a parabola?

The height equation has the form

y = a x - b x^{2}

, quadratic in

x

What is

tan θ

geometrically?

The steepness (rise over run) of the launch,

u_{y} / u_{x}