Character mistake: In his argument with Leonard regarding Superman's ability to fly, Sheldon states that in Superman 1, Lois Lane is two feet off the ground when Superman catches her as she falls from the helicopter. Not so. Superman catches Lois at least ten stories up. Additionally, both Penny and Sheldon say Superman "swoops down" to save Lois. Penny can be forgiven, but Sheldon would know better: Superman flies UP from ground level to catch Lois.
Character mistake: Sheldon is wrong about the physics of pushing the furniture up the stairs. This is not just a question of work (energy) but also of power (energy/time). Since he and Leonard cannot produce an unlimited amount of energy per second, they may need to push it slowly, contrary to what he asserts. This is a very serious and very basic error for Sheldon considering he has a PhD in Physics.
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