Discussion of the Bouncing Ball Gravity Engine.

A ball bounces up and down between floor and ceiling. Both floor and ceiling are rigid and infinitely massive. The bounces are assumed elastic, that is, the ball's velocity after impact is the same as before impact, but with reversed direction.

Now imagine that the gravitational constant g is slowly but steadily decreasing. The ball is released at rest from the ceiling. The ball attains a certain speed when it reaches the floor, and bounces back. But since g is now smaller, it still has a small velocity when it hits the ceiling. Clearly this means that on completion of this ceiling-to-floor-to-ceiling cycle it has gained kinetic energy, which we could tap with a slightly inelastic ceiling tile which would steal just that small amount of energy, bringing the ball to rest on impact. The gravitational force, though slightly smaller than before, would cause the ball to fall to the floor and bounce back to the ceiling, where we again steal the excess energy gained in this cycle, and so on until gravity disappears, or forever, whichever comes first.

Why isn't this practical?

Simple answer:

The only energy we could possibly extract from this system would be that kinetic energy the ball attains during its first fall to the floor, slightly less than mgh.

More detailed analysis:

We will analyze this by appealing only to kinematics laws and Newton's laws of dynamics, without explicit use of conservation of energy nor the laws of thermodynamics.

This analysis may be more detailed than some may find necessary, but it illustrates the thought processes one goes through when trying to figure out this puzzle.

The assumption that the ball's velocity changes only direction on impact with the massive floor or ceiling is a result of the conservation of momentum law which follows directly from Newton's laws and the assumption that the floor and ceiling are infinitely massive. The derivation is a bit subtle, but we'll assume that the reader will accept its conclusion, having seen behavior of nearly this sort when very elastic balls bounce from solid and massive concrete floors. In this case, when the gravitational constant is changing, the fact that the duration of a perfectly elastic impact is infinitesimal, and the gravitational constant doesn't change significantly during so short an interval.

The kinematics law we'll need is the relation for speed of a body under constant acceleration:

v_f² = v_i² + 2gx

The body moves from point i to point f. v_i is the speed the initial point i, while v_f is the speed the final (later) point f. g is the acceleration during this time, and x is the distance it moved. This equation assumes that g is constant during this interval.

The deliberate deception in our claim was the same as that made in the Schadewald Gravity engine. Here's the misleading statement:

The ball attains a certain speed when it reaches the floor, and bounces back. But since g is now smaller, it still has a small velocity when it hits the ceiling. Clearly this means that on completion of this ceiling-to-floor-to-ceiling cycle it has gained kinetic energy, which we could tap...

But when g is decreasing, its average value is smaller during the fall, say g₁ and therefore the ball's speed is slightly smaller when it reaches the floor than it was in the constant acceleration case. If g were to remain constant at the value g₁ (the average value it had during the fall), the ball would then only have sufficient rebound speed to send it back to a height h. But g has decreased and is still decreasing, so its average value is smaller during the rise, allowing the ball to rise higher by a small distance x. This means that when it reaches height h the ball still has a small velocity.

If we place a slightly inelastic pad at the ceiling, it can tap off a small amount of energy, slowing the ball to a stop. Then the ball begins its second cycle with zero speed.

However, the speed at the floor during the second cycle is less than it was in the first cycle, and so on, steadily decreasing during successive cycles. The period (the length of time for each cycle) steadily increases. Eventually, when g reaches zero, the ball's speed at the top and everywhere else is zero also.

But how much energy have we tapped? No more than mg₁h, the kinetic energy it had when it first reached bottom. This will not do anything to solve our energy crisis.

This is a satisfying and plausible result. When g reaches zero, there's no accelerating force so the ball must have constant velocity then. It's possible that the ball might have a small residual speed when g reaches zero. This would be a consideration if g abruptly went to zero, or if it decreased so rapidly that it reached zero during the first few cycles. In that case the ball would be left bouncing between floor and ceiling with a constant nonzero speed, but a speed less than it attained during the first half cycle. So there's no energy to tap here, energy conservation is working properly. This is the reason Bob Schadewald, in explaining his version of this deception, said: "The weight may pick up speed at the top, but never at the bottom...". This speed increase will occur if g reached zero during a cycle. The same is true of our bouncing ball engine.

If, for example, the speed suddenly dropped to zero at the end of the first half cycle when the ball reached the floor, it would rebound and hit the ceiling with the same velocity it had attained while falling. This points out the difficulty we'd have analyzing the gravity shield engine. Discontinuous changes of acceleration represent infinite values of what physicists call the jerk, the rate of change of acceleration. Analyzing such discontinuous changes mathematically requires special care, and a good knowledge of calculus. It's all too easy to blunder and reach absurd conclusions. [I'm speaking from long experience reaching absurd conclusions.]

None of these gravity machines would work any better if g were increasing.

This analysis is directly relevant to the Schadewald Gravity Engine. The deception is the same, and the actual behavior of the SGE is the same with respect to top and bottom velocities.

Related Puzzles:

1. How would steadily reducing gravity affect a simple pendulum? Amplitude? Period? Velocity at bottom of swing? Height of swing?
2. How would steadily reducing gravity affect the motion of a mass suspended from a Hookian spring (obeying Hooke's law) and bouncing up and down?

1. The period increases. The speed at the bottom decreases from cycle to cycle. The height at the extremes of the swing remains the same. The kinetic energy of the pendulum steadily decreases.