Figure about 1/2 HP or 350W for your 15 – 20 mph on level ground and smooth surface, assuming an efficient drive mechanism like a chain (not a friction wheel), and typical bicycle frontal area. A little more for wide, low pressure tires and upright rider, less for narrow, high pressure tires and reclining driver.

For verification, look at electric bicycles, this is their average power for that speed.

Note that acceleration is pretty slow if you don’t have more power than this! Also, may need gearing to get full power over range of speeds, especially while accelerating.

Edit:
Force diagrams make little or no sense for power calculations.

The bearings are quite low friction, essentially negligible. The tires will give significant rolling resistance, especially if wide and low pressure. Racing tires will slice rolling resistance dramatically. Since the spindle or axle is the only contact point for the wheel to the chassis, the force obviously acts through the axle, but also has a rotating component. Caculating actual forces is both difficult and of no practical value; structural loads far exceed loads from friction, and the forces are not useful in determining power.

At speed, most friction is due to aerodynamic drag. Find an online calculator to help. Determine the frontal area, estimate the Cd from the shape, then use the air density to calculate the drag. For rolling friction, find an online calcualtor that includes rolling friction for bicycles with different types of tires.

Take the output of the motor, derate by transmission losses, typically 2% to 10%. Determine drag/friction force as function of velocity, then solve for velocity in:
Power = force * velocity.

Determine acceleration from F = ma, where F is the excess force. Excess force is the drive force from the reflected power of the motor through the drive chain using P = F * V (at the motor rotational velocity for that speed) less the drag. This remaining force accelerates the cart. Integrating the acceleration yields the estimated response.

Finally, check the torque and power curves of the motor, reflecting them through the power transmission mechanism, to assure the motor is producing max power at max velocity, and the power train delivers sufficient torque to operate the go cart over the entire speed range. Operating at a lower power point for the motor at top go cart speed will result in unexpectedly low final velocity.

PS, sorry for spelling, Yahoo refuses to spell check.

Then you can’t consider friction zero, as in a real world situation, most of the power is used to overcome friction.

To move at a fixed speed with no friction and no air resistance, takes zero power.

What is the speed and weight?

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