Step-by-Step Problem-Solving Framework

Step 1: Identify the sub-topic

• Forces / G → Newton's law
• g varying → altitude/depth/latitude formulas
• Orbits/periods → Kepler's laws
• Energy questions → satellite energy formulas
• Escape → v_e formula

Step 2: Write the correct formula

• Don't mix up PE, KE, and E formulas
• Check for negative signs (PE is negative)
• Use GM = $gR^{2}$ to eliminate G and M

Step 3: Substitute and solve with units

• Always include units at each step
• Check dimensional consistency
• Keep 2–3 significant figures

Step 4: Verify with checks

• For satellite energy: |PE| should equal 2×KE
• For orbit comparison: higher orbit should have longer period
• For escape velocity: v_e should equal √2 × v_{0} at same radius

Calculator-Free Computation Tricks

Trick 1: Simplify fractions inside square roots v_{0} at r = 4R: v_{0} = √($gR^{2}$/4R) = √(gR/4) = √(gR)/2 = v_{0}(surface)/2

Trick 2: Use T ∝ r^(3/2) ratio directly If r increases by factor n: T increases by n^(3/2) Common values: 4^(3/2) = 8; 9^(3/2) = 27; 8^(3/2) = 16√2

Trick 3: Recognise g/n problems At altitude: (R+h) = R√n → h = R(√n − 1) At depth: d = R(1 − 1/n)

Trick 4: Planet scaling v_e scales as √(M/R): faster for heavier or smaller planets