The Explosion Scenario

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An inelastic collision and an explosion are for all intents and purpose polar opposites.   One begins with two or more objects that unite to form a single object.  The other begins as a single object that breaks into two or more pieces.  Explosions are typically violent and occur so quickly that analyzing them is difficult / messy.   However, a version of the explosion scenario occurs everytime someone throws an object providing us with the means to see why momentum cannot represent mechanical energy.

The Astronaut Wrench Experiment

This experiment can easily be replicated exactly as stated here and is representative of what occurs during any explosion but in a far more controlled manner. 

Newton’s 3rd Law is often quoted as, “For every action there is an equal and opposite reaction.” This tells us that force acts in two directions at the same time; push on something and that something pushes back with equal force. This is particularly evident in space where the effects of gravity and other factors cannot obscure one of Newton’s most remarkable discoveries.

astronautIf an astronaut during a spacewalk throws a wrench, both he and it accelerate away from one another.  Neither the wrench nor the astronaut experiences force unless the other one does as well; time is therefore the defining factor when it comes to force.  If the astronaut is more massive than the wrench, the wrench will accelerate more quickly. Newton’s 2nd Law, usually written as F=ma, explains this. Because both accelerate for the same amount of time, the less massive wrench will windup moving faster than the more massive astronaut.  Consequently, the wrench moves a greater distance than the astronaut.  And in case there is any question, distances would be measured begining from the point the astronaut was holding the wrench. 

With the Work Energy Theorem deemed a fact, the wrench will wind up with more kinetic energy than the astronaut; this is not viewed as a problem but it should be. Both the astronaut and wrench receive identical actions for the same amount of time and yet one has more mechanical energy than the other. This could be thought of as a violation of Newton’s 3rd Law even though that Law was developed specifically for motion, not energy as it is understood today.

In this simple experiment we also find the reason that momentum, the product of mass and velocity, cannot represent mechanical energy. When we take the momentum of the astronaut and add that to the wrench’s momentum, the mathematical result is always zero. Before the experiment, neither is moving and thus neither has mechanical energy nor momenta. Using the idea that the mass of the astronaut is 100m with the wrench having a mass of m, consider what happens after throwing the wrench. The wrench achieves velocity v moving to the right while the astronaut retreats in the opposite direction. Since his mass is one hundred times as great, the magnitude of his velocity will be 1/100th that of the wrench. This is because when he accelerates, he does so at 1/100th the rate the wrench does (Newton’s 2nd Law) and they accelerate for the same amount of time. We then have the following.

The wrench moves to the right and has momentum while the astronaut moves to the left and also has momentum.  Their respective momenta is exactly the same but when added together, the result is zero — no net momenta.  Mathematically, this is as follows. 

Momentum

Arrows were placed above the variables v to indicate direction; we could have just as easily used a plus sign (+), if you like, indicating movement to the right as positive with a minus sign (-) for movement to the left. In any event, the sum is as shown — zero. This tells us that none of the biological energy used by the astronaut manifests as mechanical energy if using momentum to describe energy. This is not what anyone would expect of a mathematical representation for mechanical energy. One should expect that if there were no mechanical energy and some was added, the result would be a positive value, not zero.  This means that momentum, being a directional expression is worthless when it comes to describing mechanical energy.

Interestingly, if we were to use the product of mass and speed (ms) to describe mechanical energy, an entirely different result occurs.  And in that case, the amount of biological energy the astronaut converted into mechanical would be positive.  In that case, the ms of the wrench added to the ms (100m x s/100 = ms) of the astronaut would equal 2ms (ms +ms), not zero.  Moreover, if he used a greater amount of biological energy so that the wrench traveled at a faster speed, the amount of mechanical energy would be greater just as it should be.  

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