Projectile Motion with Air Drag — Testing the 45° Rule

Source

Simulation: Projectile Motion with Air Drag — Why 45° Is Wrong (three models on screen at once: vacuum / linear drag / quadratic drag; 🚀 Launch fires a single shot; range-vs-angle curve with marked optimum).
Concept: "45° gives maximum range" is a vacuum-only theorem: with air drag the optimum drops below 45° and the trajectory becomes asymmetric (steeper descent than ascent).
Serves: Projectile kinematics; effect of resistive forces; idealization vs realistic model; reading an optimum off a function plot.
Inquiry: Predict the best angle with drag on, then find it by experiment (θ-slider sweep with a Best-so-far tracker) and compare with the ★ optimum marked on the range-vs-angle curve.
Misconception: "45° is always the best angle" — carried by prediction choice 🅐; its rebuttal appears only when that choice is selected.
Verified anchors: g = 9.81 m/s². Vacuum, v₀ = 20 m/s, θ = 45°: R = 40.77 m, T = 2.88 s. Linear drag b = 0.3 kg/s, m = 1 kg, same launch: R = 25.26 m, optimum θ* = 39°, terminal velocity mg/b = 32.7 m/s. Quadratic drag c = 0.015 kg/m, m = 1 kg, same launch: R = 28.27 m, T = 2.59 s, optimum θ* = 42°, terminal velocity √(mg/c) = 25.6 m/s. Lecture defaults (v₀ = 29 m/s, θ = 45°, m = 1 kg): b = 0.30 kg/s and c = 0.016 kg/m are paired so linear and quadratic ranges nearly match (~44.9 m).

Pedagogy principles applied

ID	Principle	How it shows up here
G1	Playable from load	All four sliders, the Vacuum / With-drag model buttons, 🚀 Launch and Play work with zero clicks; one flight plays on load then stops; prediction is optional and never gates anything.
CF	Counterfactual mode	All three models (vacuum exact, linear-drag closed form, quadratic RK4) can fly together every shot — independent visibility checkboxes; every visible path is solid. Readout focus toggles among the selected models.
H1/H2	Misconception is structural	"45° always" is prediction choice 🅐, not a standing text card; its rebuttal (with live computed numbers) renders only on selection.
C1/C2	Readout discipline	Readouts panel shows Range, Best so far, flight time, terminal speed, and decay time τ — all used by the experiment steps.
P1	Prediction first, no spoilers	The range-vs-angle curve with the ★ optimum stays hidden until the student opens 📈; the field animation alone never reveals the best angle.
E1/J5	Honest animation, stable frames	Real-second flights from exact closed-form kinematics (no integrator) for vacuum and linear drag; quadratic drag is the one numerically integrated model (RK4); the launch scene uses a fixed world window (95 m × 38 m, sized to v₀ ≤ 30 m/s) so slider changes visibly resize the arc; R(θ) plot R-axis likewise fixed at 0–95 m.
F1	Function-plot discipline	The curve marker is static and bound to the θ slider — never a self-moving dot; the ★ optimum is computed, not animated.
X1	Exact claims + audit hook	All feedback numbers are computed live from the sim's own solver; window.__audit exposes vacuumRange, dragRange, θ*, and v_T for machine verification.
A1	One concept	"With air drag, maximum range needs less than 45°."

🎯 Projectile Motion with Air Drag

Teach me Hide Text

Speed

Guided Inquiry

1 · Watch the race

Let’s start with a race. Press 🚀 Launch (or just watch the opening shot): all three models fly together — vacuum, linear drag, and quad drag — as solid paths you can toggle on and off. Notice that with drag the descent is steeper than the ascent — an asymmetry the vacuum parabola never shows — and the range is shorter too.

2 · Predict the best angle

Now for the big question. With air drag on (b > 0), which launch angle θ do you think gives the longest shot on flat ground?

3 · Hunt the peak

Let’s test that against the data. Sweep the θ slider and watch Range, while Best so far keeps your record. Near the top the peak is flat, so bracket it: does 35° beat 45°? Does 40°?

Of those three, which range do you find is largest?

4 · Read it on the curve

Now you can see the whole picture. The Range vs angle plot is open: ● is your θ, and ★ is the computed optimum. At these defaults the ★ sits near 36° — clearly below the 45° reference line.

5 · What moves the ★

So what moves the ★? At stronger drag it sits even lower. Try raising b further (or raise m and match b) and watch it slide — the flight depends on b and m only through b/m, so b = 0.30, m = 2.0 kg flies identically to b = 0.15, m = 1.0 kg. Then set b = 0 and watch the ★ snap back to 45°. The "45° rule" is a vacuum-only theorem (here drag is linear, F = −b·v; real quadratic drag gives the same conclusions).

6 · A faster shot

One last experiment. Hold b and m fixed and raise the launch speed v₀ from 18 to 30 m/s. What do you think happens to the best angle?

Listen

Controls

▾

Trajectories

Vacuum Linear drag Quadratic drag

Launch angle θ45°

30°60°

Launch speed v₀29 m/s

1830 m/s

Drag b (kg/s)0.30 kg/s

00.35 kg/s

Drag c (kg/m)0.016 kg/m

00.018 kg/m

Mass m1.0 kg

1.02.0 kg

Show axes & grid

Readouts

▾

Readout focus

Flight time—

Terminal speed v_T—

Decay time τ = m/b—

FORMAL LAYER — SI units, g = 9.81 m/s², y measured upward, launch & landing at equal height. Toggle off anytime (∑ Formal).

⚠ Vacuum: a symmetric parabola — time up equals time down, descent mirrors ascent.

sin 2θ peaks at 2θ = 90°. This — and only this — is where the "45° rule" is a theorem.

✓ Linear drag. τ = m/b is the velocity-decay time; v_T is the terminal speed of a free fall.

The exact closed form this sim flies for linear drag. Note x(t) can never pass v₀ₓτ — drag caps horizontal reach.

✓ Quadratic drag. No closed form — integrated numerically (RK4, dt = 2 ms); c → 0 recovers the vacuum solution exactly.

Flight time and range are solved on the model solution — no fitted curves. The ★ optimum is the best whole degree (matching the 1° slider).

The Physics

Projectile motion with air drag · what this sim shows, the physics behind it, and every control

At a glance

The rule that 45° gives the longest shot is derived for a ball with no air around it. This sim flies one launch three ways, in vacuum, with linear drag and with quadratic drag, so you can test whether that rule survives air resistance.

What you'll learn

How drag changes the shape of a trajectory, not just its length.
How the best launch angle depends on air resistance, launch speed and mass.
Why the linear drag law has an exact solution, and what it says about horizontal reach.
How the linear and quadratic drag laws differ, and which one real air follows.

In flight, two forces act: the weight mg, straight down, and drag, always opposite to the velocity v.

What's on screen

Vacuum (cyan): gravity only, no air.
Linear drag (amber): drag force −b·v, proportional to speed.
Quadratic drag (violet): drag force −c|v|v, proportional to speed squared.
Flags and apex line: where each ball lands, and a dotted drop from the top of the focused path.
Readouts: range, flight time and more, for the model picked in Readout focus.
📈 Range vs angle (opens beside the scene): range against angle, with ● at your angle and ★ at the best one.

See itOne launch at 45° and 29 m/s, flown by all three models. Watch for: the amber and violet balls landing about halfway to the cyan flag.

The controls

Press ⌖ on a row to see where that control sits in the sim.

Control	What it changes	Range
Trajectories	Shows or hides each model's path. Hiding a drag model also hides its slider.	3 switches all on
🚀 Launch	Fires one shot at the current settings. Every visible model flies at once.	button
Launch angle θ	The angle above the ground.	30°–60° opens at 45°
Launch speed v₀	The speed the ball leaves at.	18–30 m/s opens at 29
Drag b	Strength of linear drag, F = −b·v. At 0 the amber path is the vacuum path.	0–0.35 kg/s opens at 0.30
Drag c	Strength of quadratic drag, F = −c\|v\|v. At 0 the violet path is the vacuum path.	0–0.018 kg/m opens at 0.016
Mass m	The ball's mass. It changes both drag paths, not the vacuum one.	1.0–2.0 kg opens at 1.0
📈 Range vs angle	Opens the plot of range against angle (5°–85°), with ● at your θ and ★ at the best whole degree.	on / off
Show axes & grid	Draws metre axes and a grid over the scene.	on / off opens off
Readout focus	Picks which model the readouts, the apex line and the ★ describe.	3 buttons opens on Linear
Readouts	Range, best so far, flight time, terminal speed v_T, and τ = m/b (linear only).	panel
▶ Play / ⏸ Pause	Runs or freezes the flight. After landing, Play replays it.	top bar
Speed	Plays the flight faster or slower on screen. The physics stays the same.	0.25×–4×
↻ Reset	Restores the defaults and restarts the mode you are in.	top bar

Fixed: g = 9.81 m/s², flat ground, launch and landing at the same height, no wind and no spin.

Drag breaks the symmetry

In vacuum only gravity acts. The path is a parabola: the fall mirrors the climb, and both take the same time. Linear drag adds a force that always opposes the velocity:

τ = m/b is how long drag takes to slow the ball; v_T = mg/b is the speed a falling ball settles at.

Drag takes away speed for the whole flight, so the ball is slower on the way down than on the way up.
The slower descent covers less ground but takes longer: at the defaults, 26.3 m and 1.62 s up, then 18.5 m and 1.94 s down.
The ball leaves at 45° but lands at about 64° below the horizontal. The range falls from 85.7 m to 44.9 m.

The default launch (θ = 45°, v₀ = 29 m/s, m = 1 kg). At these defaults the two drag laws give nearly the same path.

See itThe default launch, with the readouts on the linear-drag ball. Watch for: the dotted apex line on the amber path standing right of the middle of its flight, so the fall is shorter and steeper than the climb.

The best launch angle

In vacuum the range is a theorem with a clean peak:

sin 2θ is largest at 2θ = 90°. This is the only place the 45° rule comes from.

Drag removes speed every second the ball is in the air. Steeper shots stay up longest, so they lose the most.
A flatter shot spends less time aloft, and that outweighs its lower vacuum range. At the defaults the best whole degree is 36°: 46.6 m, against 44.9 m at 45°.
The peak is flat: 35° gives 46.6 m and 40° gives 46.3 m. A small error in angle costs little.

Range against angle at the defaults. ★ marks the linear-drag best, 36°.

See itOpen the range-vs-angle plot at the defaults. Watch for: the ★ on the amber curve sitting left of the dashed 45° line, labelled best 36°.

What moves the best angle

The more drag matters, the further the best angle falls below 45°. With linear drag, m = 1 kg:

change	best angle
b = 0 (vacuum)	45°
b = 0.10 kg/s	41°
b = 0.30 kg/s (default)	36°
v₀ = 18 m/s, b = 0.30	39°

Stronger drag pulls the best angle down.
A faster launch meets more drag, since the force grows with speed. The best angle falls from 39° at 18 m/s to 36° at 29–30 m/s.
A heavier ball is slowed less by the same force, so it moves the best angle back toward 45°.
At b = 0 the vacuum result returns exactly: a symmetric parabola and a best angle of 45°.

See itStart at v₀ = 18 m/s with the plot open, then drag the launch speed up to 30 m/s. Watch for: the ★ label stepping from best 39° down to best 36°.

See itSet b = 0 with the plot open. Watch for: the amber path lying on the cyan parabola, and the ★ at best 45°.

The exact linear-drag flight

Linear drag acts on each direction separately, so the motion can be solved exactly. The sim flies this solution:

The linear-drag path, with v_0x = v₀ cos θ and v_0y = v₀ sin θ.

A wall in x. The horizontal speed decays as e^−t/τ and never reverses, so x can never pass v_0xτ. At the defaults that limit is 68.4 m; the ball lands at 44.9 m.
Only b/m matters. b and m enter only through τ = m/b, so b = 0.30 kg/s with m = 2.0 kg flies exactly like b = 0.15 kg/s with m = 1.0 kg.

θ v₀ b m

These sliders move the sim too.

See itFly b = 0.15 kg/s with m = 1.0 kg, then b = 0.30 kg/s with m = 2.0 kg, both at 45°. Watch for: both runs reading the same linear-drag range.

Real air: the quadratic law

Quadratic drag has no exact solution, so the sim integrates it step by step. At c = 0 it is the vacuum path.

For a ball moving at these speeds, air drag grows close to the square of the speed. The linear law is the simpler model that can be solved exactly.
Each law caps the speed of a falling ball. At the defaults: mg/b = 32.7 m/s (linear) and √(mg/c) = 24.8 m/s (quadratic).
The conclusions survive: the path is lopsided, the range shrinks, and the best angle sits below 45°. At the defaults it is 40° for quadratic drag.

Speed of a ball dropped from rest, for both laws at the defaults.

See itSwitch the readouts to the quadratic model and open the plot. Watch for: the ★ moving to the violet curve, labelled best 40°, still left of the 45° line.

Where it matters

Batted balls. In baseball the longest hits leave the bat at roughly 25°–35°, well under 45°. Drag, and the lift from backspin, reshape the path.
Raindrops. Quadratic drag caps their speed: a drop 2 mm across falls at about 6.5 m/s, however high the cloud.
Skydivers. Falling belly-down, a skydiver levels off near 55 m/s (about 200 km/h) within roughly 12 s of jumping.

Go deeper: which drag law applies when

The ratio of the two forces is set by the Reynolds number, Re = ρvD/η. Linear (Stokes) drag rules when Re ≪ 1: fog droplets, dust, pollen. Quadratic drag rules for balls, people and cars.
For quadratic drag, c = ½ρC_dA. With air (ρ = 1.2 kg/m³) and a sphere (C_d ≈ 0.47), the sim's c = 0.016 kg/m is a ball about 13 cm in radius, roughly basketball-sized, but with m = 1 kg a heavy one.

Summary

In one line. 45° is the best launch angle only in vacuum. Drag makes the path lopsided and moves the best angle lower the more drag matters.

Key points

Drag always opposes the velocity, so the fall is shorter, steeper and slower than the climb.
Drag costs speed for every second aloft, so flatter shots win: 36° at the defaults with linear drag, 40° with quadratic.
Linear drag depends on b and m only through τ = m/b, and caps horizontal reach at v_0xτ.

Common mix-ups

"45° is always the best angle." Only in vacuum, on level ground.
"Throw higher to win back the distance drag takes." Extra height means extra time aloft, which is exactly what drag charges for.
"Mass doesn't matter, as Galileo showed." That is a vacuum result. With air, a heavier ball of the same size flies closer to the vacuum path.

Things to try on your own

Set b = 0.10 kg/s, sweep θ to find the best angle, then check the ★.
Raise m to 2.0 kg at b = 0.30 kg/s and watch the ★ climb back toward 45°.
At the defaults, compare the Flight time readout for Linear and Quad focus.