Irodov Problem 1.230

May 19, 2026

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Problem Statement

A body is projected vertically upward from the Earth’s surface with speed v < v_escape. Find the maximum height h reached.

Given

v < v_e, M_E, R_E.

Concepts & Formulas

Energy conservation: ½mv² − GM_Em/R_E = −GM_Em/(R_E+h). Solve for h.

Step-by-Step Solution

Step 1: ½mv² = GM_Em(1/R_E − 1/(R_E+h)) = GM_Em·h/(R_E(R_E+h)).
Step 2: v²R_E(R_E+h) = 2GM_E·h = 2gR_E²·h.
Step 3: v²R_E + v²h = 2gR_Eh → h(2gR_E − v²) = v²R_E.
Step 4: h = v²R_E/(2gR_E − v²).

Worked Calculation

h = v²R_E/(2gR_E − v²).

Boxed Answer

h = v²R_E / (2gR_E − v²)

Physical Interpretation

As v → v_e = √(2gR_E), denominator → 0 and h → ∞, confirming escape. For small v: h ≈ v²/(2g), recovering the flat-Earth result.