Foundations — Range, max height, time of flight — all derived

This page assumes nothing. Before you can read "$T=\frac{2u\sin \theta }{g}$" you must know what each of those marks is, what picture it stands for, and why the topic can't do without it. We build them one at a time, each resting on the one before.

0. The picture everything lives in

Every symbol we define points at something in one diagram: a ball launched from flat ground, flying in an arc, landing again. Keep this picture in your head — we will label pieces of it one by one.

Look at the red arc: that is the path (the "trajectory") of the ball. The dashed lines are the two motions we will separate. Everything below is a name for some feature of this picture.

1. The two axes: horizontal and vertical

Picture: In the figure, $x$ runs left-to-right along the ground, $y$ runs bottom-to-top like a ladder.

Why the topic needs it: Gravity pulls only downward. So the up-down motion behaves completely differently from the side-to-side motion. Splitting the world into these two perpendicular directions lets us treat each with its own simple rule. This split is the whole trick — see Projectile Motion — components & independence.

2. Speed and velocity — $u$ and $v$

Picture: In the figure, the short red arrow at the launch point is $u$ — its length is the speed, its tilt is the direction.

Why the topic needs it: The ball starts with one push. Everything after — how long, how high, how far — is decided by how hard ($u$) and which way ($\theta$) that push was. We use the letter $u$ for the initial value and $v$ for a current value so we never confuse "start" with "now."

3. The launch angle $\theta$

Picture: In the figure it's the wedge-shaped opening between the ground and the red launch arrow.

• $\theta =0^{\circ }$ → thrown flat, straight along the ground.
• $\theta =90^{\circ }$ → thrown straight up.
• $\theta =45^{\circ }$ → exactly halfway, a "corner" tilt.

Why the topic needs it: The same speed $u$ produces wildly different flights depending on the tilt. A flat throw ($\theta$ small) goes far but low; a steep throw ($\theta$ large) goes high but not far. $\theta$ is the knob that trades one for the other.

4. Splitting the arrow: components $u_x$ and $u_y$

Here is the single most important move in the whole topic. We take the tilted launch arrow $u$ and ask: how much of it points sideways, and how much points up?

Picture: In the figure the red slanted arrow is $u$. Drop a straight line down to the ground: the shadow along the ground is $u_x$, the height it reaches is $u_y$. Together $u_x$ (bottom) and $u_y$ (side) form a right triangle with $u$ as the slanted long side.

Why sine and cosine — and not something else

We have a right triangle. Its longest side (the hypotenuse, opposite the right angle) is $u$. The angle $\theta$ sits at the launch corner. We need to turn "length $u$ and angle $\theta$" into the two shorter sides. That is exactly the job trigonometry was invented for.

The adjacent side (touching $\theta$) is the horizontal one, and the opposite side (across from $\theta$) is the vertical one. So:

$u_x=u\cos \theta ,u_y=u\sin \theta$

Why the topic needs it: Once $u$ is split into $u_x$ and $u_y$, we can treat the flat drift and the up-down flight as two separate 1D problems. See Vectors — resolving into components.

5. Acceleration and gravity — $a$ and $g$

Picture: In the figure, gravity is the fat downward arrow on the ball — always pointing straight down, always the same size, at every point of the arc.

Sign convention: We pick up as the positive direction. Since gravity pulls down, its acceleration is $a_y=-g$ (negative). Horizontally there's no force, so $a_x=0$.

Why the topic needs it: Gravity is the only thing that curves the path. It is what makes the ball slow, stop, and fall back. Without $g$ the ball would fly off in a straight line forever. Deep dive: Free Fall & Acceleration due to gravity.

6. Displacement and time — $y$, $x$, $t$

Picture: Pick any point on the red arc. Its horizontal distance from the start is $x$; its height is $y$; the moment it's there is $t$.

Why the topic needs it: $t$ is the glue. The vertical motion decides when things happen (when it peaks, when it lands); we then feed that same $t$ into the horizontal motion to find where. This shared clock is the entire coupling between the two independent motions.

7. The 1D motion equations we'll reuse

Because each axis is a plain 1D motion, we borrow two standard results from Equations of Motion (1D kinematics). Along the vertical axis (with $u_y=u\sin \theta$ and $a_y=-g$):

Why two different equations? They answer different questions:

• The first has $t$ in it — use it when you care about timing (time of flight).
• The second has no $t$ — use it when you know a velocity and want a height but don't yet know the time. At the top, $v_y=0$ is known but the time isn't, so the second tool is perfect for max height.

Choosing the right tool for the question being asked is a skill the parent note leans on constantly.

8. The double-angle identity $2\sin \theta \cos \theta =\sin 2\theta$

When we compute range we get a $u\cos \theta$ multiplied by a $u\sin \theta$, producing the combo $2\sin \theta \cos \theta$. There's a tidy identity that folds it into one term:

Why the topic needs it: It reveals at a glance that range is largest at $\theta =45^{\circ }$ (because $\sin 2\theta$ peaks when $2\theta =90^{\circ }$) and that complementary angles share a range. Without it the formula still works but hides its own secret. Full treatment: Trigonometric identities — double angle.

How these foundations feed the topic

Read top to bottom: axes, speed, angle, and trigonometry all combine into the components; the components plus gravity and the shared clock feed the 1D equations; those equations (helped by the double-angle identity) generate the three boxed results in Range, max height, time of flight — all derived.

Equipment checklist

Test yourself — cover the right side and answer aloud.