Problems 24 through 34 deal with a mass–spring–dashpot system having position function x(t) satisfying Eq. (4). We write x₀ = x(0) and v₀ = x'(0) and recall that

The system is critically damped, overdamped, or underdamped, as specified in each problem.

Deduce from Problem 24 that the mass passes through x = 0 at some instant t > 0 if and only if x₀ and v₀ + px₀ have opposite signs.

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p = c/(2m), w = k/m, and w=w2 - p².