To calculate the torque needed for a 24-inch single wheel vehicle weighing 150 lbs to reach 25 mph in 5 seconds, the required acceleration is approximately 7.04 ft/sec². Using this acceleration, the torque at the wheel is estimated to be around 35 ft-lbs, with a recommended motor size of at least 5 hp to account for inefficiencies. The discussion also highlights the complexity of using a hydraulic drive system versus a direct engine drive, with considerations for remote control and variable speed. Concerns about stability at higher speeds are raised, suggesting a maximum speed closer to 5 mph for safety. Overall, the focus remains on determining accurate torque and horsepower values for the vehicle's design.