How to build a joule mobile that will travel 10 meters?

My understanding is that a joule mobile is one that makes use of 1 Joule of energy (1W.s) with the goal of moving as far as possible. You have chosen an extreme goal, so an all out effort is needed. I don't know if you have to use home made parts or that you can use better wheels and pulleys from elsewhere. You should collect the parts and draw it all out in detail before you start, so you don't make any mistakes and have to start again.. I have never seen one of these, so I am making this all up.

The normal source of energy is potential energy - e.g. 1kg mass falling through 0.1m with Gravity = 9.8m/s/s is 0.98J (Pe=mgh).

I don't know of you can consider other energy forms, for example a flywheel, spring, or a charged capacitor and electric motor. By way of example energy stored in a capacitor is 1/2(CV^2) so one Joule is 2000uF charged to 10V.

The basic problem whichever way you go is that 1J is not much energy. The important thing therefore is to absolutely minimise friction (and other losses) to maximise distance travelled.

An all out approach is needed:

1) The body of the carriage needs to have a clean shape, and be long an wide enough to be stable with the load it has to carry. It needs to be strong and stable enough to carry the load without flexing. Balsa wood with braces comes to mind. It could be a rectangular box with the wheels/axles near the top to lower the center of gravity. The size could be 6 inches wide, 10 inches long, 2 inches deep.

2) The overall weight needs to be low to minimise friction.

3) The wheels need particular attention. These suffer from rolling resistance where they contact the surface they run on. A larger diameter helps, and probably a reasonably narrow wheel helps - more like a bicycle wheel than a car wheel. (see first link). Hard wheels are better than softer rubber tired ones. Consider the surface they will run on - dirt and gummy stuff will increase the friction - sweep it clean first.

I think a diameter of three inches is about right. Think about two screw top jar lids fastened together back to back for each wheel. The problem is to find the centers exactly.

4) The wheel bearings and axles need special attention, so the wheels will run true and not wobble or run in different directions from each other. They need to be carefully aligned. The wheels can be fixed to axles, but the axles need to be mounted on something that rotates very freely, even with the loads you expect. They should be stiff. Plastic or metal tubes come to mind, like electrical conduit. Miniature ball bearings suit the needs best. Alignment and centering of all moving parts is important.

The carriage should run very easily, so that it will run on a flat table with the slightest tilt. Think that it will probably need to coast most of the distance at a slow speed.

The drive. For a dropping weight, a tower is needed. It has two legs, one each side and maybe a front/back support too (guy wire?). I would make that from balsa wood too, maybe located in the middle or towards the front of the carriage over the drive wheels. Calculate the weight to suit the height. I think about 333mm is right, so around 300g. Allow for the pully diameter. The weight has a string that goes over a free running pulley at the top of the tower, and then to the front axle. It is wound around the axle like a windlass, so that the pull rotates the axle and wheels as it falls. The string should be long enough that the weight hits the bottom just as the string runs out and pulls out of the hole in the axle, leaving it all free to coast. I don't suppose you can use the energy in the weight when it hits the bottom? In theory you should have used it all up.

Edited later...

It looks like basic division. 300 meters divided by 10 sec. = 30 meters per second toward the South.