Many physics students perform an impressive experiment that proves the Work Energy Theorem. It is so impressive in fact that no one ever stops to analyze its basic premise. There is no arguing with the results and yet, it contains a hidden flaw so insidous that discovering it requires suspecting the unthinkable. In other words, if you believe the Work Energy Theorem is valid to the slightest degree, it is nearly impossible to discover the experiment's flawed premise. Despite this, it is relatively simple to expose if a willingness to learn something new about something old exists.
The following graphic shows the usual experimental setup. The premise of the basic experiment or any version thereof is both simple and ingenius. Because the object M2 can fall, it produces a force that pulls the object M1 via the pulley. Object M1 is on an air track that allows it to move horizontally with minimal resistance. And by the use of various sensors strategically placed (not shown), this setup allows students to test the Work Energy Theorem.
This is a clever experiment; the object M2 begins near the pulley and has potential energy while M1 begins at rest and has no kinetic energy. As M2 falls, its potential energy vanishes and as that occurs, M1 accelerates thereby increasing its kinetic energy. This experiment actually demonstrates the relationship between gravitational potential and kinetic energy but in a slightly convoluted manner.
A similar experiment can be done without the table, air track, and pulley. It is described below as the “Generic Version” and represents what happens when using a single object. Think of dropping an object and measuring the necessary variables using advanced remote sensors. In essence, the Academic Experiment and the Generic Version are basically identical in that both can test the relationship between potential and kinetic energy but, the Generic Version does so more directly. Like many technical things, the actual means to accomplish something can sometimes appear more complicated than the theory behind it. In this case, the seven steps of the Generic Version are the basic theory behind the experiments physics instructors use regardless of any embellishments they might add.
Generic Version
Drop an object of known mass (m) from a known height and measure its velocity (v) as it reaches the ground.
The force (F) on the object is found by the formula, F=ma=mg.
The magnitude of the displacement (distance) is equal to the known height.
Multiply the amount of force determined above with the displacement. In other words, determine the work done to the object (Work = force × displacement).
Take the velocity of the object as it reaches the ground and its mass and plug those two values into the kinetic energy formula (ke=½ mv^{2})
Compare the energy calculated in Step 4 with the value determined in Step 5.
Repeat Steps 1 through 6 from a height twice as great.
When any version of the academic experiment is done, the results are extremely impressive. Unless a gross error occurred, the values calculated for work and kinetic energy always match within the expected margin of error. Moreover, when Step 7 is performed thereby doubling the work done, the kinetic energy of the object doubles. In short, this experiment appears to make an indisputable case for the Work Energy Theorem. Unfortunately, all such experiments contain a fatal flaw. The flaw is not obvious given how physics is taught. While the experiment actually verifies the mathematics of the Work Energy Theorem, it does not conclusively prove that mechanical energy must be quantified by Fd or ½ mv^{2}. This becomes immediately apparent when doing the exact same experiment with two very slight differences. This is illustrated in the “Alternate Version”.
Alternate Version
A. This is identical to Step 1 of the Generic Version. Drop an object of known mass (m) from a known height and measure its velocity (v) as it reaches the ground.
B. Measure the time (t) it takes for the object to reach the ground.
C. Same as Step 2 of the Generic Version. The force (F) on the object is equal to F=ma=mg.
D. Multiply the force determined in the previous step with the time the object falls to determine the total impulse (Ft) applied to the object.
E. Plug the velocity of the object as it reaches the ground and the object’s mass into the formula for momentum (p=mv). (The variable “p” is often used to denote momentum.)
F. Compare the values of impulse (Ft) and momentum (mv).
G. Repeat Steps A through F from a height four times as high.
When this version of the same basic experiment is done, the results are just impressive as before. Unless a gross error occurred, the values calculated for impulse and momentum always match within the expected margin of error. Moreover, when Step G is performed thereby doubling the time force acts, the object’s momentum doubles. This version also appears to make an indisputable case but this time for a different equation. This version is never used to prove the impulse / momentum equation is scientifically valid and, it is definitely not used to show that Ft and mv should represent mechanical energy.
As has been stated elsewhere on this website (the Explosion Scenario), momentum cannot represent mechanical energy. And once again as per the Scientific Method, that fact does not elevate Fd and ½ mv^{2} to the status of a proven scientific fact. Mathematically, these two ideas do equate to one another which is the only thing the Academic Experiment proves. If we use non-directional variables and repeat the Alternate Version of the Academic Experiment, we find that ft = ms (force x time = mass x speed). In other words, the Academic Experiment is worthless when it comes to determining the mathematics that should represent mechanical energy. It proves that Fd = ½ mv^{2} , Ft = mv, and that ft =ms but, not which of these three equations we should be using when talking about mechanical energy.
The Bottom Line
The only valuable thing the Academic Experiment does is verify the mathematics of three different but related equations. Unfortunatly, two of the three equations are never considered. This is because teachers (who were once students) miss the fact that whenever force acts and an object accelerates, two related factors change. One of these is distance; nothing can change its velocity or speed without traveling across some distance. Likewise, time also changes; nothing can instaneously change its velocity or speed. Interestingly, both factors are mentioned in the motor sport of drag racing. Two cars begin at rest, accelerate through a distance of a quartermile and the one that does so in the shortest amount of time is the winner. Such races could also be done where the winner is the one that covers the greatest distance for a given amount of time. In other words, time and distance are not the same thing but they are related to one another AND both must be considered whenever dealing with motion.
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