A \(20 \mathrm{~g}\) ball is fired horizontally with initial speed \(v_{\mathrm{i}}\) toward a \(100 \mathrm{~g}\) ball that is hanging motionless from a \(1.0-\mathrm{m}\)-long string. The balls undergo a head-on, perfectly elastic collision, after which the \(100 \mathrm{~g}\) ball swings out to a maximum angle \(\theta_{\max }=50^{\circ}\). What was \(v_{\mathrm{i}}\) ?