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My request for input is this: If I add the extra battery (I’ll need to find a place to stick it) without changing to a 60v controller, is there a chance it will work for a period of time and then burn out? What can I expect? Some controllers can handle being overvolted and some can not. It may work indefinitely, it may immediately burn out or, or it may work for a period of time and then burn out.

If the controller must be changed, does the controller and the motor need to be switched together as a team? No they do not. You can  install a controller with a higher Voltage than the motor. The motor may run hotter than normal if a big enough load is put on it, so you would want to watch the temperature of the motor if you over Volt it.

How much and which product do you recommend if I switch to a 60v controller and motor. Is the conversion fairly straight forward? Are there any special tools required?  The controller depends on how many Watts the motor is and visa versa. I recommend using a controller and motor with the same Volts and Watts rating. Installing a different controller and motor could range from easy with no special tools needed, to difficult with modifications to the frame being necessary depending on the controller and motor that are being installed. 

Whether I do the volt increase or not, I am mostly interested in allowing this bike to move past the 25km/h pre-set limit. Can you guide me to a solution to increasing the speed of this bike?  It sounds like you are on the right path. I would start with a bigger battery pack and controller and take it from there.

Out of curiosity, how would modifying the 48v controller to a faster allowable speed change the over-all range of the bike AND how would changing the voltage to 60v change the over-all range.  The range of the bike depends on how many Amp hours the battery pack is rated for and how many Amps the motor and controller are drawing from the battery pack. The faster the vehicle goes the more power it will draw and the lower the range will be. I could not say for sure what the results of the modification will do the the range. You would have to try the modification first and then see what the new range is.

Do you have any idea what the top speed of this bike would likely be in both voltages if there were no speed restriction on the controller?  No I do not. I would expect an increase in top speed if you overvolted the controller and motor though.

Unless your controller has a built in function to do so, I do not know of any way to modify or adjust a controller to make the motor go faster. Chances are that your existing controller does not have a speed limiter feature, but that the speed is being limited by the Voltage of the power source and maximum output Amperage parameter of the controller, thus the output Voltage and Amperage of the controller.

Our brushless speed controllers do not have speed limiters or adjustments to limit or increase the motors speed. Some of our brushed speed controllers do have power level adjustments though. With our speed controllers the motors speed will be limited only by the Voltage and Amperage that it receives from the controller. They do not have built in speed governors.

Brushless motor and controller systems often feel like they have speed governors because the motor will not turn faster than a certain RPM, however the speed is being governed by the Voltage of the power source and not the controller or motor.

Changing to a higher gear ratio will increase the top speed of the scooter if the motor has enough power to propel the vehicle in the new higher gear. If the motor has enough power the higher gear will increase the speed, however if the motor is not powerful enough then a higher gear will make the scooter go slower.

For example a car with a 200 horsepower engine going uphill in 4th gear will start to slow down, but a car with a 350 horsepower engine will have enough power in 4th gear to accelerate up the same hill.