Consider a spring subject to a frictional or damping force.
(a) In the critically damped case, the motion is given by x = c₁e^rt + c₂te^rt. Show that the graph of x crosses the t-axis whenever c₁ and c₂ have opposite signs.
(b) In the overdamped case, the motion is given by x = c₁e^r1t + c₂te^r2t. where r₁ > r₂. Determine a condition on the relative magnitudes of c₁ and c₂ under which the graph of x crosses the t-axis at a positive value of t.