I am now Ridemonkey's expert on electric bicycles because I just built one yesterday and test rode it.
One thing about 31mph on the motor vs 51mph pedaling is how the crankset works in conjunction with the motor.
The bike I build had a freewheeling crankset. So you can pedal while the motor is working or not.
However the bike is still driven by the chainring and the rear cogs. So even if the motor is assisting the pedaler (not the other way around), you would need to turn a 53/11 gear ratio at almost a 150 cadence to go 50 mph.