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We study the ballistic pendulum in a situation that is not usually considered where the bullet shot into a hanging block requires a finite time (the collision time) to come to rest relative to the block. During this time the block undergoes a non-negligible displacement. A slow collision is one for which the block's displacement during the collision time is an appreciable fraction of its maximum displacement. For slow collisions, the quantitative results of our model differ significantly from the standard ballistic pendulum treatment. Suitably generalized momentum and energy relations are evaluated and provide a check on the numerical solutions. This work extends the treatment of a familiar problem and demonstrates the utility of computational tools accessible to undergraduates. [ABSTRACT FROM AUTHOR]