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As the question states, what is our current best machine for translating falling gravitational potential energy, such as a large weight, into launching a smaller projectile vertically? A lever? A trebuchet? What sorts of efficiency levels have been achieved?

What I'm looking for is the best way to translate:

$$ g * h * m_{\textrm{weight}} * \textrm{efficiency} = \frac{1}{2} * m_{\textrm{projectile}} * v^2. $$

Edit

Here's an example: I want to launch a 10kg mass at 100m/s using only gravitational energy (no chemical rockets, no rail guns, etc.). What's our most efficient machine to do this, and what percentage of the energy of the falling weight will actually be transferred to the projectile? This would get me an idea of the size of the weight and how much it has to fall to accomplish this. The only machine I know of that does this on a large scale and for decent size masses is a trebuchet, so this might be a good starting point unless there are others I'm unaware of.

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A cliff? Or a vertical tube in a vacuum –  Martin Beckett Sep 12 '12 at 14:38
    
Those don't transfer the energy from a falling weight into launching a projectile... –  Ehryk Sep 12 '12 at 17:52
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They do transfer it into ke of the falling object - that's all you asked for! –  Martin Beckett Sep 12 '12 at 18:23
    
The suspicion that he wasn't answering the question you meant is probably why Martin offered that as a comment rather than a "answer", but it really is the indisputably correct answer to the question you asked. You'll want to think carefully about what your requirements are in order to phrase the question you really want to ask. Probably best to edit this one rather than making a new post. –  dmckee Sep 12 '12 at 19:47
    
I thought $m_{weight}$ and $m_{projectile}$ made it clear that they were different masses. Does the edit ask the question that I really want better? –  Ehryk Sep 12 '12 at 21:25

3 Answers 3

up vote 3 down vote accepted

This question was asked four months ago, but none of the existing answers mentions trebuchets.

To my knowledge the trebuchet design is the only design that is purely mechanical. Other projectile throwing devices store elastic energy and on release transfer that to the projectile.

So a trebuchet it is.
(In effect 'lever' and 'trebuchet' are synonymous. A trebuchet gets its leverage by being a lever (pun intended)).

It may be worthwhile to research the noble art of pumpkin chunkin'. There's a town in the US where there is a yearly pumpkin chunkin' contest that also features a trebuchet category.

Can a trebuchet deliver close to 100% efficency?
For one thing, frictional losses can be kept quite low, that's good.
The problem is this: to throw the projectile the lever arm must be accelerated to a large angular velocity. After the projectile has been released the lever arm is swinging violently. That kinetic energy of the lever arm is energy from the counterweight that was not transferred to the projectile.

The trebuchet design involves trade-offs. A longer lever arm gives the potential for faster throws, but a longer arm also has a bigger moment of inertia.

An ideal trebuchet would transfer all of its kinetic energy to the projectile, so that right after releasing the projectile it would just sit there, nearly motionless. I'm not aware of any trebuchet design that achieves that.

To reduce the inefficiency the mass of the lever arm must be as low as possible. That is what the pumpkin chunkers do: they make the lever arm as flimsy as they dare. For every throw they stand well back, as their machines tend to self-destruct when the trigger is pulled.

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A perfectly efficienct trebuchet would be still when the projectile is launched. In fact most of the energy goes into kicking the trebuchet backwards. Experimental proof involves a large model and a couple of crushed foot bones –  Martin Beckett Dec 29 '12 at 17:47
    
But what would be a good estimate for the percentage of energy transferred to the projectile? –  fibonatic Aug 4 at 11:20
    
The caveat being that the elasticity of the structure absorbs energy so the bigger it is the less efficient it is, unless it is made of masive structural members. So in the end there is balance between efficiency and cost/size. –  ja72 Aug 4 at 13:25
    
@Ehryk - You might want to take a look at algobeautytreb.com/trebmath356.pdf for the math behind treb simulators. It's germane since his sims indicate a maximum efficiency of about 80%. –  WhatRoughBeast Aug 5 at 0:42

I vote for a pulley. Anchor the pulley at near the height of the "main weight" and it can transfer the falling vertical energy nearly completely (a factor of elasticity of the rope or cable). Move the anchor point up or down to control launch angle.

For a inelastic cable, efficiency a factor only of its capacity to bend.

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Incorrect on the "nearly completely" bit. The weight difference of the two masses gets to accelerate both masses, and in fact dumps most of it into the greater mass (presumably the one that's not the "projectile"). –  genneth Sep 12 '12 at 21:36
    
Huh? The only way there would be loss is if the rope was elastic. –  Mark J Sep 12 '12 at 21:41
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I'm not talking about loss. I'm saying that there's a lot of energy left in the "weight", and not all the initial potential energy is converted to the "projectile" motion. –  genneth Sep 12 '12 at 21:42
    
Right, this may translate pe to ke in the weight, but not much of the falling weight's energy is translated into the launch of a projectile. –  Ehryk Sep 12 '12 at 22:01
    
Eh? If the length of the rope is about the height of the weight (+ pi*r of the pulley wheel) keeping the pulley anchored at that same height, then nearly all of it is translated to the projectile. To keep with energy conservation, the mass of the falling weight gets converted to greater velocity of the (lighter) projectile.... –  Mark J Sep 12 '12 at 22:14

Have an insulated cylinder of gas with a piston. Put the weight on the piston and let it press down. Then lock the piston in place and put the other mass on. Then turn the cylinder on its side and unlock the piston and the other mass will be accelerated. By turning it on the side, all the energy goes into accelerating the second mass and not into lifting it. The mass of the piston needs to be small in comparison to the second mass, because it is being accelerated also. But any device will have that problem. To get a vertical direction as requested, then have a curved ramp to convert the horizontal motion to vertical. The size of the curved ramp will consume some of the energy also.

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It sounds like this would require large, long tubes with tight clearances, and the pressurization of the gas would (under extreme conditions) heat the gas high enough to melt the tube. Government agencies have used a similar idea to launch scramjet engines, and run into the temperature limit often; as does QuickLaunch, Inc. I truly want a mechanical design that can be scaled up to 200,000kg dropped 91m being translated into a 10kg mass lauched at 6km/s. I don't think the temperature inside the cylinder of this design can be scaled to this. –  Ehryk Sep 13 '12 at 0:20
    
In considering this design though, couldn't the drop tube be set to $90\deg$ and launch tube set to the desired angle? Would it be able to use any sort of hydraulic advantage by having the launch tube at, say, 1/2 the diameter so it moves twice as fast? –  Ehryk Sep 13 '12 at 0:23
    
@Ehryk - Hold on just a minute. First is was 100 m/sec, and now it's 6 km/sec. You might want to rethink this. "First the bucket, then the pail, then the laboratory scale. Ever bigger, ever faster. Faster, faster, then - disaster." –  WhatRoughBeast Aug 5 at 0:40

protected by Qmechanic Aug 4 at 10:22

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