Wow. . . Just wow
May I suggest a nitro RC car?
If you are hell bent on electric then get some high quality LiIon batteries. They are half the weight, last 5x longer and are much more efficient. They stay at full charge longer then die very quickly at the end of the charge cycle, unline NiCad which just slowly die linearly. The weight decrease alone will help you go further and longer.
Solar panels are trickle type, they do not have "current on demand" for things like a DC motor. It is possible, your idea, but you would need to park the car in the sun for a few hours to recharge the LiIon cells (using approx 1 sq ft panel.) Just having hours of continous driving, which is what I believe you are after, is not possibile with todays solar technology.