There’s a lot of good stuff here. Most people seemed to get the right answer (so far as I can tell), although they used different paths to reach it. My vote goes to shabadu because he sums it up so succinctly (brevity is always a virtue), he writes clearly, he mentions a couple details that other posters missed, he gets everything right (so far as I can tell), and he adds a note of humor at the end. I regret that he didn’t attempt the calculation regarding the first vehicle that Theodire Minick included, but Minick seems to use some unstated assumptions that bother me. For instance, he assumes that the lead balls have potential energy based on a 4-foot difference, but how did he come up with that number? It has to be a guess, and therefore doesn’t justify his calculation to umpteen significant figures!
I don’t think any of the posters took into account the likely behavior of the balls on the roof. When the vehicle accelerates (assuming it does) the balls in the channel will tend to roll backward relative to the vehicle, because of their inertia. Therefore they will roll off the channel at increasing speed relative to the vehicle, and therefore they will hit the chute lower down, with greater kinetic energy, creating a more powerful forward thrust. On the other hand, their rearward motion will cause them to hit the lower chute at an increasing angle, which will deliver less forward thrust.
Of course the balls will not roll along the upper channel with zero friction, especially if they tend to rub against each other; and the balls in the reservoir over the cab will be significantly constrained. Therefore their mass, in addition to the mass of the vehicle, must be overcome by the forward thrust. However, the aggregate mass will diminish as the number of balls diminishes, and this will enable the vehicle to accelerate faster. Since this is a complex system involving factors such as the friction of lead against lead and the precise contouring of the ball containment system, no precise calculations are possible.
A simple way to look at the first vehicle is to assess the energy which would be required to lift the balls up onto the roof in the first place. This is the most energy that you can get back out of the system by allowing the balls to drop. In practice you will get less, because of frictional losses everywhere in the system.
Another issue which was not addressed is the question of “where the energy goes” in vehicle number two. The motor, after all, is doing work, circulating the water. If that work is not translated into motion, what happens to it? The answer of course is that it is converted to heat by friction between the water and the pipes. I would expect the water to become perceptibly warm as the truck sits there churning the water around and around while going nowhere.
And here’s Shabadu answer…
In the first vehicle, the falling balls will apply a forward force. Just like pool balls bouncing off a bumper, the falling ball will be deflected backward by their impact with the second trough. According to Newton’s third law of motion (“To every action there is an equal and opposite reaction.”) the vehicle will also deflect forward from this same impact. In an ideal system this will move the vehicle forward. However, any deformation of the second trough or the tires, and friction in the axles, and even wind resistance, will very likely overpower any forward force applied by the falling balls.
The second vehicle won’t move at all. The same principle that applied forward force to the first vehicle exists in the second one, so the water hitting the trough does give it a forward force, but since the water is collected it hits the back of the trough/pipe when it turns at the bottom to return to the pump. This impact applies a second force in the opposite direction of the first. In addition, the frictions, deformations, and wind resistance that might completely stop the first vehicle would be even more detrimental in this vehicle because the water will probably have less momentum than the lead balls, which provides less force to be transferred into movement of the vehicle.
On the other hand, the second vehicle does have a better chance of moving after the police arrest the first driver for dropping huge lead balls all over the road.