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Drawing 100 Amps from a 7ah is possible, in an ideal case a fully charged 7Ah battery would deliver 100amps before being depleted for 70 seconds. IE Time = Capacity/current drawn T=7000/100 = 70Seconds.
The key in calculating battery life is to use the watts you know for the calculation of the amps at the battery voltage. You are way out on fuse ratings as the opening current you refer would cause nuisance tripping, However, As above a fuse opening time is also related to the current it receives and time period it passes, rupture is normally many times greater than the actual rating it is given and denoted by Time lag, slo blow, fast acting etc for example the circuit may employ a 15 or 20amp fuse but a thermal overal overload rated at 5 amps. Rating is all design dependant.
With respect to the differing motors you refer all of this is taught in an academic qualification – Stairlift Technology, Output RPM is worked out from torque required and PCD of Rack and pinion. Your opening calculation law is also used widely in calculating power ratings. As mentioned earlier the actual running current is much less than you think, more like as I have posted, less than a couple of Amps, start up current is very brief. As above I suggested running times and 10 journeys. A simple calculation of working out the average watts consumed over 20 starts (10 full journeysâ€™) and watts used with the lift not charging and in standby for the remaining duration of the 24 hours would be a simpler calculation to confirm power required against power used. Design factors of safety are also fully explained in the course.