Two parallel senses depending on which
弹 you mean: read tán with
起 it means 'spring/bounce up' (a ball, a spring) OR 'start playing' (guitar, piano, pipa — instruments you
弹 by plucking/striking). Read dàn (different tone) it would mean 'bullet' — but
弹起 in this 'spring up' compound is always tán.