I came up with this question while thinking about pulleys. Most textbook pulley problems involve the force on the load being equal to the load's weight so that there's no acceleration. But what if we had the following setup. There's an anchor on the ceiling and there's a heavy crate on the floor attached to a moveable pulley and then a fixed pulley on the ceiling near the anchor and they're all connected by a steel cable that we'll assume is strong enough to handle all tensions without breaking or stretching and light enough to be considered mass less. Rather than just pull down on the cable and enjoy a mechanical advantage of two, I build some scaffolding and put a heavy crate on top of the scaffolding and connect the steel cable to that. Then I push the crate off of the scaffolding but the steel cable has some slack so the crate is free falling before the cable becomes taut and starts to exert a force on the other crate. Lets assume that the second crate is heavy enough or falls far enough that the initial force exercted on the first crate is enough to get it moving rather than the second crate just bouncing on the end of the cable. Just eyeballing it I think they're will be acceleration and time-varying tension on the cable but I don't really know how to write the differential equation.