I ran some gear ratio calculations for this project using our MOT-24350G 24 Volt 350 Watt 420 RPM gear motor which has a 9 tooth sprocket along with a 44 tooth sprocket for the wheel. I calculated the 48" wheel circumference to be around a 15" wheel diameter. Here is the calculation.
This combination of parts provides a 3.84 MPH top speed which should be equal to a fast paced walking speed.
It sounds like the cart will be driven around 30 to 60 minutes per day. For this amount of drive time a 24 Volt 15Ah battery pack should be sufficient however it looks like this cart will be used at races or commercially so a battery pack that has power to spare such as a 24 Volt 22Ah battery pack might be better suited for it. For an 8 hour or less recharge time a 24V 1.6 Amp to 2 Amp charger could be used with a 15Ah battery pack, or a 24 Volt 3 Amp to 4 Amp charger with a 22Ah battery pack.
I can recommend other parts needed for this project such as a speed controller and throttle, or I could make a kit for this project if you prefer.
I made a kit for your project with a set of two 22Ah batteries. It is item # KIT-146 and is available on this page:
If you would like I can make the same kit except with two 15Ah batteries and a smaller battery charger which would lower the price of the price of the kit by around $50.00.
Here is a photo of the parts included with KIT-146. What is not shown in the photo below is the battery pack wiring harness and all of the connectors that we attach to the parts so they are ready to plug together as soon as you receive the kit.