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Projectile Range Hoodie

45 DEGREES GOES FARTHEST | Pixelated Physics

\[R = \frac{v^2 \sin 2\theta}{g}\]
$54.99 USD
$54.99 USD
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Size guide S–5XL · inches

What launch angle throws farthest? On level ground with no air resistance, the range is R = v² sin 2θ / g. Because sin 2θ peaks at 90°, a 45° launch goes farthest, reaching v²/g. Angles that add up to 90°, such as 30° and 60°, land at the same spot. Galileo described this in 1638. Black pixel-art hoodie; equation, symbols and sources in the physics panel below.

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Details

Size & fitS–5XL

Unisex Heavy Blend hoodie (Gildan 18500), classic fit.

SizeWidthLengthSleeve
S202733.5
M222834.5
L242935.5
XL263036.5
2XL283137.5
3XL303238.5
4XL323339.5
5XL343440.5

Measurements in inches. Width and length ±1 in, sleeve (from center back) ±0.75 in.

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Delivery times & where we ship

Returns & replacements21 days

Every item is printed to order, so we can’t accept returns or exchanges for change of mind, or if you ordered the wrong size or colour.

Print defect, misprint, damage or the wrong item? We’ll send a free replacement, or a refund if you prefer. Report it within 21 days of delivery with a photo; no need to send it back.

Full refund & replacement policy

The physics

Without air resistance, a projectile's range is v² sin 2θ / g, which is largest at 45 degrees. All trajectories at a given launch speed stay under a parabola-shaped envelope.

Throw a ball at the same speed every time, with no air resistance, and only the angle changes where it lands.

On level ground the range is R = v² sin 2θ / g. sin 2θ is largest at 2θ = 90°, so 45° goes farthest, with R = v²/g. Because sin 2θ = sin(180° − 2θ), angles that add up to 90° land in the same place: 30° with 60°, and 15° with 75°.

Every possible throw at that speed stays under one curve, the safety parabola y = v²/2g − gx²/2v². Each trajectory touches it at exactly one point.

On the front, the five arcs are 15°, 30°, 45°, 60° and 75°. The 45° arc is amber and lands farthest; the 30° and 60° arcs share one landing disc and the 15° and 75° arcs share another. The filled dome is the safety parabola.

Galileo showed in 1638 that, without air, a projectile follows a parabola and 45° gives the greatest range.

Equations

\[R = \frac{v^2 \sin 2\theta}{g}\]
\[R_{\max} = \frac{v^2}{g}\ \text{at}\ \theta = 45^\circ\]
\[y_{\text{env}} = \frac{v^2}{2g} - \frac{g x^2}{2 v^2}\]
SymbolMeaningUnit
\(R\) horizontal range on level ground \(\mathrm{m}\)
\(v\) launch speed \(\mathrm{m/s}\)
\(\theta\) launch angle above the horizontal \(\mathrm{rad}\)
\(g\) gravitational acceleration \(\mathrm{m/s^2}\)
\(x\) horizontal distance from the launch point \(\mathrm{m}\)
\(y_{\text{env}}\) height of the envelope (safety parabola) at x \(\mathrm{m}\)

Sources

  1. Halliday, Resnick & Walker, Fundamentals of Physics, Ch. 4: Projectile Motion
  2. Taylor, Classical Mechanics (2005), Sec. 2.3 (projectile with no air resistance)
  3. Galileo (1638) Discorsi e dimostrazioni matematiche intorno a due nuove scienze, Fourth Day

Source Galilei, G. (1638). Discorsi e dimostrazioni matematiche intorno a due nuove scienze, Fourth Day (motion of projectiles). Leiden: Elzevir.

Field notes

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