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Using v² = u² - 2gh, where final velocity v = 0, u = 20 m/s, g = 10 m/s². Therefore, 0 = 400 - 20h, giving h = 20 m.

Velocity is a vector quantity with both magnitude and direction, unlike speed which is scalar. Velocity can be negative, and in circular motion, velocity constantly changes direction.

Young's modulus is a material property independent of the wire's dimensions or elastic deformation state. It depends only on the material composition and atomic structure, not on physical changes.

At critical angle θ_c, refracted ray becomes parallel to the interface. Beyond this angle, no refraction occurs; light completely reflects back into the denser medium (total internal reflection). This principle enables fiber optics.

Carnot efficiency: η = 1 - (T_cold/T_hot) = 1 - (300/400) = 1 - 0.75 = 0.25 = 25%. This is the theoretical maximum for any heat engine.

Electric field at surface: E = kQ/r² = (9 × 10⁹ × 8 × 10⁻⁶)/(0.02)² = (72)/(4 × 10⁻⁴) = 1.8 × 10⁷ N/C.

Wavelength λ = v/f = 1500/(50 × 10³) = 1500/50000 = 0.03 m = 3 cm. Higher frequencies in denser mediums produce shorter wavelengths.

Maximum velocity in SHM: v_max = ωA = 2πfA = 2π × 2 × 5 = 20π cm/s ≈ 62.8 cm/s.

After 30 years = 3 half-lives: N = N₀ × (1/2)³ = N₀/8. The sample reduces by half three times successively.

Stars appear as point sources. Turbulent layers in the atmosphere refract starlight randomly, causing intensity variations (twinkling). Planets appear as extended sources, so refraction averaging effect minimizes visible twinkling.