Re: Energy Conversion and Newton's Laws

Marcus:

You've asked a very good question because it highlights the difference between energy and linear 
momentum.

Let's assume for a moment that you are in a rocket which is out in space far away from any large 
gravitating body, and that there are no large outside forces acting on your rocket. Now, you start 
your rocket engine and you note that your rocket accelerates as long as your engine is on.

Most rockets work by combining a fuel and a source of oxygen, then igniting the combination. 
Some of the chemical potential energy stored in the molecules is converted into kinetic energy of 
the gaseous products of the chemical reactions. Some of the molecules escape directly into space 
and do nothing to accelerate the rocket.  The remaining molecules hit the engine nozzle, and get 
reflect into a backward direction.  It is these collisions which cause the rocket to move in the 
forward direction.  

As you correctly pointed out, during the collision between a gas molecule and the rocket nozzle, 
Newton's Third Law says that the force on the molecule by the nozzle (causing the molecule to 
exit out the back end) is exactly matched by the force on the nozzle by the molecule (causing the 
nozzle to move forward).  In principle, we can add up all of the force acting on the nozzle by all 
of the molecules to figure out how much the rocket accelerates.  

However, there is a  much easier way of finding out the speed of the rocket at any time during its 
acceleration, and that is using a special form of Newton's Second Law called the Conservation of 
(Linear) Momentum: in the absence of an external force, the linear momentum of a system of 
particles does not change.  The linear momentum of a single particle of mass (m) is (m)*(v), 
where (v) is the velocity of the particle.  Conserving the linear momentum means that the sum of 
all of the momenta never changes.  This means that if the rocket was already moving forward 
with some original linear momentum, then when hot gas leaves the back end at some speed then 
the rocket must speed up in the forward direction such when you add the (negative) linear 
momentum of the exhaust gas to the higher (positive) linear momentum of the rocket, you end 
up with the original value of the linear momentum you started with.

The key point here is that you must throw some mass "away" if you wish to change the velocity of 
your rocket.  Since the gas is moving it has kinetic energy, and you asked how much of the 
propellent's original chemical potential energy is "lost" in the form of kinetic energy of the 
exhaust gas. This "lost" energy is not really "wasted," since you have to eject mass in order for 
the rocket to work. However, we can ask how to maximize the energy efficiency of the rocket, or 
equivalently, "what is the smallest fraction of the original chemical potential energy which is 
turned into kinetic energy of the exhaust gases?"

Any introductory physics textbook will show that the final speed of the rocket depends on three 
parameters:  the original mass of the rocket plus propellent, the final mass of the rocket, and 
how fast the gas is ejected out of the back end.  Your rocket can reach its final speed either by 
throwing out a lot of mass at a low speed or by throwing out a little bit of mass at a very high 
speed.  

Most chemical rockets (like the Space Shuttle) throw off a lot of mass at a relatively slow speed. 
New ion drives, like the one which powered the Deep Space One satellite, throw off a very small 
amount of material at very high speeds (unfortunately, the acceleration obtained from an ion 
drive is much smaller than the acceleration due to gravity on the surface of the Earth, so it 
cannot be used to launch objects from the surface of the Earth into space). 

Some simple algebra using the relationship for the final speed of the rocket and the definition of 
kinetic energy shows that the fraction of the total available kinetic energy in the rocket exhaust  
depends only on the ratio of the final and initial masses of the rocket. Most chemical rockets lose 
more than 80% of their original mass while boosting into orbit. For a rocket which is 20% of its 
original mass (80% of the mass is lost), over 90% of the kinetic energy is in the exhaust.  For an 
ion drive, where the final mass is of the rocket is 90% of the original mass (10% of the original 
mass is ejected), the exhaust gases contain less than 0.2% of the total kinetic energy!  The 
break-even point, where half of the kinetic energy is in both the rocket and exhaust gases, 
occurs when 58% of the mass is ejected (final rocket mass is 42% of the original mass).

The ion drive is clearly much more efficient in terms of converting potential energy into kinetic 
energy of the rocket. However, what you pay for in terms of conversion efficiency you lose in 
terms of acceleration - ion drives provide only very small accelerations (although they can run for 
very long periods of time).  

Acceleration is usually more important than energy conversion efficiency when designing a 
rocket. A rocket starting on the surface of the Earth must be able to maintain an upward 
acceleration greater than the downward acceleration due to gravity for at least the length of time 
it takes to get to where you want to go (although too large of an acceleration can cause 
structural failures - this is why explosions break things!). 

Ion drives are currently not capable of achieving such large accelerations. Chemical reaction-
rockets can achieve the required accelerations, but because the ejection velocity of the exhaust 
gases is relatively low, a large amount of mass must be ejected in order to achieve the necessary 
acceleration. Unfortunately, this also means that most of the original chemical potential energy is 
not converted into kinetic energy of the rocket, but is converted into kinetic energy of the 
exhaust gases instead. This energy is not "wasted" in the sense that it somehow lost due to 
inefficiencies in converting from chemical potential energy to kinetic energy. The exhaust gases 
must have most of the kinetic energy in order for the rocket to maintain its acceleration!  
Unfortunately, the fairly small amount of energy stored in chemical bonds limits the maximum 
possible ejection speed of the exhaust gases, which in turn forces us to expel a great deal of 
mass  in order to maintain the desired acceleration.  If the exhaust speed could be somehow 
greatly increased, then the amount of ejected mass would decrease, and the energy efficiency of 
the rocket would increase automatically. Note, however, that the efficiency can never be 100% - 
exhaust gases must always have some non-zero amount of kinetic energy for rockets based on 
the principle of  Conservation of Linear Momentum. 

The situation is slightly different in a gun, since the explosive gases are not being ejected out of 
the back of the gun.  Assuming that all of the kinetic energy of the gas goes into the kinetic 
energy of the bullet and the gun (not exactly true since you see a flash of light and hear a loud 
sound coming from the barrel, both of which are forms of energy), then you can show quite 
easily that the fraction of the total kinetic energy stored found in the bullet is a function only of 
the ratio of the two masses (bullet and gun). Since most guns are much more massive than the 
bullets they expel, almost all of the kinetic energy of the gas goes into the bullet and not the 
gun.  For the gun and the bullet to both have half the total kinetic energy would require that the 
gun and the bullet both have the same mass (very large bullet or very small gun!). Note that this 
result is a little different than the result for the rocket I mentioned above (half of the kinetic 
energy is in the exhaust when 58% of the original mass is ejected as exhaust and 42% or the 
mass remains in the rocket). This difference is due to the fact that the mass of the rocket slowly 
diminishes with time, so less mass has to be ejected as time goes on in order to maintain a 
constant acceleration. 
 
Some energy will be wasted when you convert from chemical potential energy to kinetic energy 
(the second law of thermodynamics guarantees this), and we can detect some of this wasted 
energy in the form of light and sound. Usually the amount of energy which is not turned into 
kinetic energy is a fairly small fraction of the total amount of converted energy.

To summarize, Newton's laws require that kinetic energy be shared between the moving objects 
(exhaust gas and rocket; gun and bullet), although not necessarily equally. We usually use the 
phrase "wasted energy" to indicate energy which is lost due to some inefficiency in how we 
transport or use it. Implicit in the term "waste" is the idea that there is some way in which we can 
collect this energy and use it again. Defined in this way, we see clearly that the kinetic energy of 
the rocket's exhaust gases and the gun's recoil is not wasted, since the sharing of kinetic energy 
is a consequence of Newton's laws of motion, and it is not possible to "collect" the kinetic energy 
of the gun's recoil or the rocket's exhaust gases and use the energy to make the bullet or rocket 
go faster.