 Solution

ID: past-exam-of-the-physics-a-course-of-the-university-of-cambridge/2019/paper-2/section-b/b7/c/solution

x (0) = 0

and

\overset{x}{˙} (0^{+}) = v_{0} > 0

x (t) = ⎩ ⎨ ⎧ \frac{v _{0}}{ω _{d}} e^{- γ t} sin (ω_{d} t), v_{0} t e^{- ω_{0} t}, \frac{v _{0}}{s} e^{- γ t} sinh (s t), γ < ω_{0}, γ = ω_{0}, γ > ω_{0}, s = γ^{2} - ω_{0}^{2} .

All curves start at the origin with the same positive slope

v_{0}

. Underdamping gives a decaying sequence of positive and negative lobes. Critical damping gives one positive lobe followed by return to zero. Overdamping also remains positive for these initial data, but has a longer tail. The initial impulse changes velocity abruptly, not displacement.

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