Character mistake: It's more than unlikely that a prison guard would leave a suspected murderer (George) alone with anyone, especially a woman (Edna) for minutes. (00:24:28)
Character mistake: When first discussing the projectile Murdoch has written on his board that 2 (sin45) =1. This is obviously incorrect to anyone who's studied even basic trigonometry. (00:05:54)
Other mistake: James Pendrick shares his plans with Murdoch about how he is going to reach the orbit of the Earth by flying his own built rocket. However, earlier in the episode, Pendrick himself reveals that he is claustrophobic (extreme fear of confined places). Having such phobia would make his planned journey fundamentally impossible as he couldn't stay hermetically closed in the small rocket for long without suffering serious psychical damage.
Other mistake: Pendrick states that the rocket barrel is 70 metres long (deep). Despite that, Murdoch reaches the bottom in mere seconds, has enough light to see around and is even able to hear the others outside.
