The Mathematics of Reactionless Drives

(The maths is at the bottom of this article.)

One of those guys?

When I was young, I wondered if gravity could be polarised. "You know, like with a magnet? North and south – on and off. Then you could just take off – without all those exhaust gases. Or maybe, you could build a machine that could go forward without a propellor, or a jet, or a rocket...like in sci-fi movies? You know..."

I wondered a lot of things in those days. I was young and, unburdened by the need for established proof, full of wonderous ideas. One of those guys!

That was quite a while ago now. Pre-internet. But that’s how it started. And, after abandoning thoughts of manipulating gravity, that's how I drifted into the field of (what I now know to be called) reactionless drives.

There are lots of those guys

I stumbled across a book in my local library. “Beyond 2001: The Laws of Physics Revolutionised”. The book is inventor Sandy Kidd’s telling of his own adventures with reactionless drives – where he builds and develops his own machine(s) to investigate the possibilities. His devices were based on spinning gyroscopes and, after extensive testing, documented as having produced reactionless thrust. Although this area had been investigated (and continues to be) by a number of people, I want to acknowledge Sandy in particular, as I know I would have lost interest long ago without his work.

The joy of possibility

For me, the joy of Sandy's book came from knowing it worked but not how it worked. I’d spend hours studying what he had written and then lie down somewhere quiet, trying to construct his machine in my mind, so that I too could attempt to work out the process by which it seemed to work. I even tried building a very crude example myself. (It worked – sort of.)

Math? Maths?

One of the repeating themes in his book was the desire for someone to formalise the process that his – and other machines – seemed to be exploiting. In other words: to do the maths. He explained in an interview, that his friend and colleague Dr Bill Ferrier, was working on the maths of the device and reported getting close. Sadly, Dr Ferrier passed away before his work was completed. (If someone is able to update/ correct me on this, I would be very grateful.)

Surely it can't work...can it?

Although an inspiration to many, the pioneers were far from universally applauded. Citing Newton’s third law of motion (to every action there is an equal and opposite reaction) detractors would quickly abandon their instinctive curiosity and – often without seeing the evidence – casually dismiss the results as misreadings.

Gather the data. Examine the data.

Like many others, I chose to put my faith in the science of the process: this machine had run and the data had been recorded. (Now that the internet has made this verification straightforward, I’m glad I did.) So here was the problem then: it worked, but any explanation for how it worked was far from conclusive.

Be Curious

How I wished I was better at maths! How wonderful it would have been to spend just a few minutes staring at an image or two and then effortlessly scribbling down the formula. Still, at least I was curious. So, I thought about it. For a long time.

I Wonder?

To be fair, it wasn’t as if that was all I did: just lie back and think. I had a normal life, with normal aspirations. It was just in my quieter moments that my mind would drift back to the notion of how a machine could propel itself forward without an equal and opposite reaction? Sometimes I’d manage to ignore it. But, sooner or later, the questions would start popping back into my head again. What if this? What if that?

Eureka!

But to what purpose? History had shown that progress in this area would be made against the tide of opinion. Even if successful, what then? Perhaps the best that could be hoped for was to be ignored. (Particularly when considering the alternative was likely to be ridicule, or even accusations of falsehood.) But plod along I did. For years. And then a thought struck me. What if the action and reaction could be imagined merely as two opposing blocks of kinetic energy? (Which they are.) And what if a part of that energy could be changed into another form? (Which it can.) And what if more of the kinetic energy on the reaction side than the action side could be transformed? (Which it can.) And what if this transformed energy could be recycled back into the system? (Which it can.) What then, eh?

It started to make sense

And it started to make sense. It made sense with ball bearings; it made sense with gyroscopes. It made sense with water, bullets, gases, shockwaves – with any system I could imagine. If only I could do the maths…

The maths wasn't that hard after all

I now have the maths to back up my theory. That is to say: I think I have the maths to back up my theory. I’ve checked it again and again, and – as far as I can tell – it respects established science and seems to offer a simple and logical explanation of what is actually going on.

The Candyfloss Experiment

Perhaps rather timidly, I initially “published” the theory in a sci-fi novel I wrote a while back. (It’s called The Candyfloss Experiment and it’s available on Amazon). I wrote the concept into the narrative and then I put the maths in as an appendix at the back – pretending that one of the story's characters had written it. I guess I was feeling the water. If it all blew up, I could slip away with the claim that it was only a work of fiction. But where’s the courage in that? After all, I had benefitted from other people’s efforts. Some of which bore fruit. Some of which didn't. But all of which helped others with their own thinking. So, surely it’s only fair that I put it out there for real?

So here it is: The Mathematics of Reactionless Drives

So here it is: Synthesising External Force by Converting Kinetic Energy Within an Enclosed Collision System. (Residual Momentum Thrust.) I hope you find it interesting. But if you think it’s a load of misguided nonsense: let me know why. I’d like to learn. If you think it contravenes one or more of the existing laws of physics: let me know which ones – and how it contravenes them. I don’t pretend for a minute to know it all. I'm learning, just like you. So, tell me. Help me to learn. But, if you are one of those guys, I hope that you can find something in it, to help you along with your own adventure.

Synthesizing External Force by Converting Kinetic Energy Within an Enclosed Collision System. (Residual Momentum Thrust.)

Tim Bateup 08/06/23

Introduction

This paper explains the principles by which a machine could propel itself without propelling mass in the opposite direction to its own direction of travel, i.e. a reactionless drive. This is explained by the equation:

√(KE2m)1 + √(KE2m)2 = √(KE2m)1′ + √(KE2m)2′ + E

(Where E is an energy form other than kinetic energy, KE)

Key principles

Notion 1: Within an enclosed collision system, if some of the kinetic energy involved in a collision is changed to a form other than kinetic energy, the total momentum after the collision would be different to the total momentum before the collision.

Notion 2: Changing a part of one side of an enclosed collision system’s kinetic energy to a form other than kinetic energy will have the same effect as an external force acting on the system.

Method

1) Expressing momentum, p, as a function of kinetic energy, KE.

KE = ½mv²

and

p = mv

By multiplying ½mv² by m/m (i.e. by 1)

KE = ½mv² x (m/m) = (1m²v²)/2m

KE = (mv)²/2m = p²/2m

=> p = √(KE2m)

2) From the law of conservation of momentum. (Total momentum after a collision will be the same as the total momentum before a collision.)

p1 + p2 = p1′ + p2′

But as p = √(KE2m), this can also be expressed as…

√(KE2m)1 + √(KE2m)2 = √(KE2m)1′ + √(KE2m)2′

From notion 1: This is only true if none of the kinetic energy, KE, changes to another form of energy, E, within the enclosed system, e.g. heat.

(As the total mass of the enclosed collision system cannot change, the only possible variable is kinetic energy, KE, which can change to another form of energy. This could be achieved by, for example, using the collision’s impact to deform one of the collision’s objects and turning some of its KE to heat.)

Therefore, in an enclosed system in which some of the kinetic energy is transformed into a form other than kinetic energy, E…

√(KE2m)1 + √(KE2m)2 = √(KE2m)1′ + √(KE2m)2′ + E

or

p1 + p2 = p1′ + p2′ + E

3) From notion 2: If more of the kinetic energy from one side of the collision than the other side is converted to a form other than kinetic energy, the total kinetic energy after the collision will now be less than the total kinetic energy before the collision. This will cause (or increase) an imbalance between the momentum of the two sides of the enclosed collision system. (And a corresponding shift in the centre of mass of the overall system.)

i.e. √(KE2m)1 + √(KE2m)2 > √(KE2m)1′ + √(KE2m)2′

or

p1 + p2 > p1′ + p2′

4) If the momentum has changed then the velocity must have also changed.

From F = ma

As a = Δv/t (The change in velocity over time.)

Then F = m Δv/t

As mass is unable to change and velocity changes with momentum: a force, F, must exist if the momentum changes.

Conclusion

Using Newton’s first law of motion. (A body will remain at rest unless an outside force acts upon it, and a body in motion at a constant velocity will remain in motion in a straight line unless acted upon by a straight line force.)

As the collision system is enclosed and the overall momentum has changed: the force has behaved identically to an external force and can, therefore, be said to have been synthesised as a result of transformation of some of the collision system’s kinetic energy.

Simple example

Two machine guns are mounted facing each other at opposite ends of a friction-free trolley (in a sealed vacuum tube). At the halfway point of the trolley is a wall. One side of the wall is made from lead and the other from plasticine. As both guns fire at equal rates the collision system will move in the direction in which the bullets are firing into the lead wall. This is because lead will deform less than the plasticine and will, therefore, lose less of its kinetic energy to heat.

On lead side of collision p1 > p1′ but on the plasticine side p2 >> p2′

Implications

1) A machine could be created to propel itself without propelling mass in the opposite direction to its direction of travel.

2) A limit to this machine’s terminal velocity would no longer exist.

3) The kinetic energy transformed within the system could be transformed into a more usable form of energy than heat. E.g. The collisions could produce electricity, which could be recycled within a repeating set of collisions.

Photo by Pixabay.