1. Energy Conservation in Free Fall • When an object falls, potential energy (PE) is converted into kinetic energy (KE). • Formula: P Einitial = KEimpact 1 mgh = mv 2 2 • Solving for velocity before impact: v 2 = 2gh p v = 2gh (1) (2) (3) (4) 2. Crater Formation and Scaling Laws • The size of the crater depends on the kinetic energy of impact. • Crater diameter D follows a power-law relationship: D ∝ (mgh)n (5) n D ∝ (KEimpact ) (6) 3. The Scaling Exponent n • n is the scaling exponent (also called the empirical scaling factor). • Determines how crater diameter depends on impact energy. • Typical values: – Sand/Granular materials: n ≈ 0.25 – Harder surfaces (rock, ice, etc.): n ≈ 0.3 – Planetary-scale cratering: n ≈ 0.28 − 0.33 • Since energy spreads in 3D, crater diameter increases less than linearly with energy. 4. Experimental Design for Cratering Study • Independent Variable: Height of the drop (h). • Dependent Variable: Crater diameter (D). • Control Variables: – Same ball (mass, size, material). – Same surface material (e.g., sand). – Same measurement method (camera/ruler). • Expected Trend: D ∝ hn where n ≈ 0.25 (7) 5. Graphing and Data Analysis • Graph to verify theory: Plot D vs. h0.25 . • If n = 0.25, the data should form a straight line. • Can also fit data to the equation: D = khn and solve for n. 1 (8) 1. What is Radioactive Decay? • A process where an unstable atomic nucleus loses energy by emitting radiation. • Occurs randomly but follows a predictable statistical pattern. • Types of radioactive decay: – Alpha decay (α) – Emission of a helium nucleus (42 He). – Beta decay (β − or β + ) – Conversion of a neutron to a proton (or vice versa), emitting an electron/positron. – Gamma decay (γ) – Emission of high-energy photons. 2. Radioactive Decay Law N (t) = N0 e−λt (9) where: • N (t) = Number of undecayed nuclei at time t. • N0 = Initial number of nuclei. • λ = Decay constant (probability of decay per unit time). 3. Half-Life (t1/2 ) ln 2 λ (10) 0.693 = 346.5 s 0.002 (11) A = λN (12) A = A0 e−λt (13) t1/2 = Example: If λ = 0.002 s−1 , t1/2 = 4. Activity and Decay Rate Measured in Becquerels (Bq), where 1 Bq = 1 decay per second. 5. Energy and Radiation Effects • Alpha particles (α): Heavy and low-penetrating (stopped by paper/skin). • Beta particles (β − or β + ): Medium penetration (stopped by plastic/aluminum). • Gamma rays (γ): High-energy and highly penetrating (requires lead/concrete shielding). 6. Radioactive Dating • Used to determine the age of objects by measuring isotope decay. • Common methods: – Carbon-14 dating – Used for organic materials (half-life: 5,730 years). – Uranium-Lead dating – Used for geological samples (half-life: billions of years). 2