MOMENTUM, KINETIC ENERGY, AND THE MISUSE OF PHYSICS AND MATHEMATICS
Some days ago I was reading the debate between Gary Novak and Pierre Latour regarding the absurd hypothesis by the former, suggesting we “eliminate” the kinetic energy equation and replace it with the momentum equation.
Of course, I have no problem in totally supporting what Pierre Latour (who is a very competent and expert Chemical Engineer) wrote. His statements and calculations were totally correct, whereas those by Gary Novak were totally wrong, and no absurd claim of “corrupted science” can change that.
Therefore, my arguments below are only further evidences of the bizarre and indefensible statements by Gary Novak, maybe other readers can add their own.
Actually, the “arguments” by Novak regarding kinetic energy and momentum, are quite similar to those sophistic paradoxes from the philosopher, Zeno of Elea, in 5 B.C. when he said that the fast Achilles could not reach the turtle, because any times he moved on the road, he had to make ½ + ¼ + 1/8 of cm., and so he could not reach the turtle making the same road, because he had to run “infinite little spaces”.
The Zeno paradox was an error, of course, because the limit of ½ + ¼ + 1/8 …+ 1/n is mathematically converging to 1, and doesn’t tend to infinity, first of all. Moreover Zeno was forgetting the TIME, both Achilles and the turtle are traveling on a physical space/time = velocity, so you can always compare the velocities of both.
But Zeno was living 2,500 years ago, and NOT in 2013 as Gary Novak!
So, it is clear that Novak is confusing momentum (p = mv) with kinetic energy , which is Force x displacement (Fs) = ½ mv², when he wrote that kinetic energy should be mv (???) and not ½ mv², as any high school students know.
In doing so, Novak “forgets” that kinetic energy refers to acceleration, work and power, namely totally different concepts than momentum only.
Momentum is very important in Physics, because it is commonly used to calculate instant impulses, and energies involved in collisions, shocks, etc., such as car accidents, shootings, etc.
However, considering momentum only is useless when trying to know how much WORK (= kinetic energy) is performed by a moving body, on a limited space or distance (s), and undergoing an acceleration (a).
Novak wrote that ½ mv² was “paralleled” by Einstein to his famous equation E = mc² and that’s correct, but of course Einstein was not the first scientist using kinetic energy equation, both Leibnitz and Newton (that Gary Novak likes) two centuries before knew this very well and used concepts like force, energy, kinetic energy, etc.
Moreover, Newton was also one of the inventors (along with Leibnitz) of calculus.
So, Newton knew that ½ at² was the space (s) traveled with a velocity (v) in a time (t) and with acceleration (a), because he knew that – since v = at – if we make the integral s = ∫v dt and s = ∫ at dt, then we get:
S = a ∫t dt = a ½ t².
And so, if we multiply F = ma per s, we get:
(1)
Fs = ½ m a²t²,
and , as v = at, then a²t² = v², so
(2)
Fs = ½ mv²
that Gary Novak dislikes…
Fs means Force x displacement (space) = Work = kinetic energy.
EXPERIMENTAL EVIDENCE OF KINETIC ENERGY EQUATION
It doesn’t take a collider of subatomic particles to prove this – but only a stone, a chronometer and a bridge on a river or on a sea (taking care that no boat is passing under you).
Now, Fs can be re-written as: mas , and a (acceleration) is = g (gravitational acceleration).
Then, if you take your stone, having a mass = m under gravitational acceleration g, and let it drop and splash from the bridge into the river, then we can re-write our kinetic equation as follows:
F s = m a s =
m g s = ½ m v² = ½ m a²t² = ½ m g²t²
(3)
mgs = ½ m g²t²
then m simplifies with m and disappears, and g with g², thus we finally get:
(4)
s = ½ g t²
or
(5)
t = √(2s/g)
And those are the famous equations of falling bodies, that were discovered by Galileo at the end of 1500, proving that the velocity of falling bodies is not influenced by their masses, but ONLY by gravity (g) and air resistance.
If you remove the air, in a vacuum chamber, and let a feather drop with a hammer, or any other heavy objects, then you can see that they reach the soil at the same time, as Capt. David Scott too showed on the Moon and also demonstrated to TV viewers: (‘Mythbusters’) – ‘Galileo’s Hypothesis Hammer and Feather Drop.’
And that’s another experimental evidence showing that Mathematics, Physics and experiments are perfectly matching, provided you make the correct and precise calculations, and use the correct laws of Physics. And at last you can let your stone (not so much influenced by the air resistance) drop, and take the time of fall.
Then if it takes – say – 3.6”, then you make : 3.6² = 12.96. (12.96 * 9.8) /2 = 63.5 meters, more or less, and you can be sure that you’re above the Golden Gate, in San Francisco. Or maybe you found that your stone “splashed” in 2.89”, and so 2.89² = 8.36 . (8.36 * 9.8)/2 = 41 meters, and you were on the Brooklyn bridge, in NY.
Therefore, sorry for Gary Novak, but any times we calculate the distance traveled by a falling body, having ONLY the duration of fall (or having just space without knowing time), we are CONSTANTLY using the kinetic energy equation and the law of falling bodies, that is another form of writing the kinetic energy equation, and – sadly for him – we have to square the time to find the correct solution.
THE ROCKET AND THE ENERGY
Another confusion, and whimsical calculation by Novak was the one of rocket. Novak compared a rocket, undergoing a negative force from its own weight, and a loss of fuel mass, while traveling, with the simple situation of 2 falling bodies (1 having a mass 4 times bigger than the other), undergoing only gravity (+ air resistance), to work out a bizarre calculation regarding the alleged – in his mind – “huge” energy that the “corrupt” Physics would miscalculate for the rocket.
Sorry for Gary Novak, but maybe he didn’t realize that thousand rockets have successfully been launched , after 1940, and maybe (maybe) that was accomplished because “corrupted” physicists and engineers have correctly calculated the energy rockets needed.
It is totally wrong, for instance, what Novak wrote about the energy involved in rockets traveling.
Correct rocket equation, is as follows:
(6)
m *dv/dt – Fext = ux* |dm/dt|
where:
m* dv/dt = momentum
Fext = rocket weight (under gravitation near earth surface)
ux = velocity of expelled gases
dm/dt = loss of rocket mass due to fuel consumption
So, if we integrate the equation above when rocket is in outer space, undergoing no gravitation and Fext, we get
(7)
vf mi
∫dv = ux ∫dm/m
vi mf
namely:
(8)
Vf– Vi = ux * ln (mi /mf )
Where mf is the final mass of rocket (without fuel), and mi the initial mass (full of fuel), and vf is the final velocity and vi is the initial velocity.
Let’s make a practical calculation.
If we have a rocket whose mass without fuel is 2,000 kg. and whose mass with fuel is 20,000 kg., then we have that mf/mi = 20,000/2,000 = 10
Then, if we suppose that gases are expelled with a velocity (ux) of 2 km/sec. , our final velocity of rocket would be (starting from 0):
(9)
Vf = 2 km./sec. * ln 10 = 2 * 2.3 = 4.6 Km./sec.
Now, and that’s interesting to debunk what Novak said, let’s calculate what would be the final velocity of rocket, if we increase a lot the {154653b9ea5f83bbbf00f55de12e21cba2da5b4b158a426ee0e27ae0c1b44117} of fuel content, and reduce the “initial ” full mass, so that initial mass is only 200 kg. and mass with fuel is 20,000 kg., so mi/mf = 20,000/200 = 100
Thus, we have:
(10)
Vf = 2 km./sec. * ln 100 = 2 * 4.6 = 8.6 km./sec.
Thus, even if we increase 10 times the fuel energy of our rocket, as a ratio with its own mass, then the increase of velocity will just be 2 times!!
And that happens because – as any serious engineers and physicists know very well – the logarithmic equation (8) is a strong limit to the increase of rocket velocity!
So, this is exactly the contrary of what Novak said!
Serious physicists and engineers are making calculations regarding rockets taking care mostly to the rocket speed, and they know very well that there is no fancy “huge energy” that a rocket can receive from external forces, but the actual velocity and energy of a rocket are mathematically very constrained and limited.
To summarize: the problem is not with alleged “corrupted” Physics and Mathematics.
Mathematics is commonly called “the language of Physics”, and the language on Natural Sciences (Chemistry, Engineering, etc.)
The problem arises only when someone is misusing Physics and Mathematics.
Alberto Miatello
September 5, 2013
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