$60°$ and $30°$ are complementary angles, so $\sin(2\times60°) = \sin120° = \sin60° = \sin(2\times30°)$.

Range $R = \dfrac{u^2\sin2\theta}{g}$ is the same for both. Heights and times of flight differ.

Gothic architecture (12th–16th century, primarily European) is characterised by:

• Pointed arch ✓ — the defining feature
• Flying buttress
• Ribbed vault
• Large stained glass windows

Other options:

• Barrel vault = Romanesque architecture
• Pendentive = Byzantine architecture (used to transition from square base to circular dome)
• Hypostyle hall = Ancient Egyptian architecture (hall with many columns)

Pointed arch is the most iconic and structurally significant Gothic element.

1. Heat (temperature): drives recrystallisation and mineral reactions; increases with burial depth and proximity to magma
2. Pressure: confining pressure (lithostatic) increases with depth; directed pressure (stress) causes foliation and lineation
3. Chemically active fluids: water, CO₂, and dissolved ions catalyse mineral reactions, enable mass transfer, and are essential for many metamorphic reactions

$v_P = a + 2bt$, $\quad v_Q = f - 2t$

Setting equal: $a + 2bt = f - 2t \Rightarrow 2t(b+1) = f-a \Rightarrow t = \mathbf{\dfrac{f-a}{2(1+b)}}$