please teach me about gearing and give me some advice

my top speed is barely 45 MPH, so I am torn with a decision, everything I read states that electric motors are happy at higher RPMs, so initially I was planning on replacing the rear sprocket with a 50 tooth one to give me that covetted 5:1 ratio, but then I am being told to go faster I need to lower the ratio and replace the front sprocket with a 13 tooth one while keeping the existing stock rear, which would give me a 3.5:1 ratio

are my goals of a higher speed and optimal motor performance mutually exclusive, or is there a happy medium?

Your top speed will be limited by the motors maximum continuous output
generally power required follows this formula:

Power needed = speed^3*constant + speed x constant

so if speed increases, power needed increases dramatically

say at 45mph you need 10kw,

If your motor at its optimal rpm is capable of 10kw, than the best speed you can maintain continuously is 45mph.

So, how much power are you currently using to hold 45mph?
Is that power less than the max continuous rating of your motor at its optimal rpm?

If not, than changing the ratio to go faster will mean you can go faster, however, it becomes a game of making sure the motor doesn't overheat before your battery depletes.

If you aren't presently measuring how much power you are using, I highly recommend buying a cycle analyst.

Matt

To answer why people usually change the rear sprocket instead of the front: you get more precise control over the gearing. The smaller number of teeth on a front sprocket mean that each tooth added or subtracted makes a much bigger difference in the final ratio - you may overshoot or undershoot your ideal ratio. If, OTOH, you can get the exact ratio you want by changing just the front sprocket, then by all means that's the way to go.

are my goals of a higher speed and optimal motor performance mutually exclusive, or is there a happy medium?

Assuming that you have a peak RPM for your motor based on voltage input.

If you want to optimize your bike for top speed you need to gear higher, then your motor will be turning slower at the higher speed. It will be less efficient during acceleration because with a higher gear you get more rear wheel RPM but less rear wheel torque.

If you want to optimize your bike for acceleration you need to gear lower, then your motor will spin up to speed quicker. Not only will it accelerate quicker but if you were to accelerate at the same rate as before it will be more efficient because you are producing more rear wheel torque for the same motor RPM.

Now, if you want great top speed *and* great acceleration *and* good efficiency you need to do one of a few different things:

(a) use a gearbox that allows you to select between two different ratios. Some folks have adapted motorcycle gearboxes to do this. Theoretically you most likely don't "need" more than 2 gears. In practice you have to consider the availability of gearboxes and the fact that gearboxes are easier to make reliably when there's not a large disparity between the ratios.

(b) use a clever controller that can do "field weakening" - this is where the controller can change the electromagnetic behavior of the motor (stronger magnetic fields mean more torque (for given amps) but less speed (for given volts)). Antiscab has experience with the Vectrix controller that takes this approach.

(c) use a clever motor that has multiple windings - a winding appropriate for high torque and another winding appropriate for high speed and the ability to switch between them. The latest Enertrac motor does this.

(d) better dilithium crystals...