Why is Velocity Squared?

From henk:

Glenn, thanks for your answers. I am still thinking about kinetic energy. In the 17th century, experiments by Willem Jacob 's Gravesande showed that the striking of a ball in clay was proportional to squared velocity and later on a French physician showed it was proportional to 'mass times squared velocity'. What is it about, is my question? How to get the formulae not by math manipulation, but from experimental considerations? By the way, I explained to a friend the idea of matter-motion by using examples and despite he is a dummy in math and physics, the matter-motion explanation seems to be more appropriate to grasp physics even at his age of 75.


Thanks again for the comment. Before the Gravesande experiment, one would have thought that doubling the velocity of a microcosm would result in double the impact. Not so. As you mentioned, the doubling of velocity causes four times the impact. In other words, a microcosm impacting a soft, wet clay at velocity 2v will create a crater 4 times as deep as a microcosm impacting the clay at velocity v; an auto crashing into a wall at 40 mph will suffer 4 times the damage as one crashing into a wall at 20 mph. Your question: What gives?

I am afraid that this is one occasion in which I will not be able to avoid math, as much as I would like too. The simple reason that velocity is squared is the fact that there is a macrocosm.

Motion Without a Macrocosm

This was explained by Newton’s First Law of Motion (The velocity of a body remains constant unless the body is acted upon by an external force). This observation, the law of the universe, makes Newton the most brilliant scientist who ever lived. Thus, a microcosm moving through “empty space” at 2 m/s will move two meters in one second. At a velocity of 1 m/s it will move one meter in one second. In other words, if Gravesande’s wet clay was really “empty space,” doubling the velocity would have produced a “hole” twice as deep. Of course, there would be no “hole” and the microcosm would not stop either, because empty space offers no resistance.

Motion With a Macrocosm

This was explained by Newton’s Second Law of Motion (The acceleration a of a body is parallel and directly proportional to the net force F and inversely proportional to the mass m, i.e., F = ma.). While the First Law is just an astute observation concerning inertia, this Second Law describes causality. A cause produces acceleration, that is, a change in velocity. Here, Newton is describing a cause as a “force.” Of course, “forces” do not exist, only microcosms exist. The force concept is a handy, necessary mathematization. It is especially useful when we really do not know the actual cause. The true cause of the acceleration of any particular microcosm must be at least one other microcosm that collides with it. Incidentally, this is why determinists deny the possibility of ESP. “Extra Sensory Perception” is the indeterministic hypothesis that something or some motion might be perceived without microcosms colliding with a sensory organ.

Back to velocity squared… As generally explained, the Second Law is all about increasing the velocity of microcosms (acceleration). The collider hits the collidee. The same equations explain the opposite result (deceleration) when the situation is reversed and the collidee becomes the collider and the collider becomes the collidee. In either case, we are recognizing the effect of the macrocosm—the presence of something other than empty space. The explanation below, from http://hyperphysics.phy-astr.gsu.edu/hbase/ke.html#c3, gives the standard mathematics showing how a microcosm gets its motion:  

So what does all this mean? First, remember that, in neomechanics, there is no such thing as “energy” or “energy of motion.” Energy is neither matter nor motion. Instead, we define energy as a calculation: the multiplication of a term for matter times a term for motion. Nevertheless, the calculation or “energy concept” admitted above is a handy way to help us understand matter in motion. The illustration above shows how the microcosm gets its motion even before it collides with the clay. The somewhat mysterious “force” in the illustration is simply the push provided by some other microcosm. Incidentally, indeterminists who believe in finity often speculate about where the “first” push came from. Of course, this question becomes moot for an infinite universe—there is always yet another microcosm to do the pushing.

The gist of the KE explanation is the equation: Work = KE = Fd = mad = ½mv2. If you have ever pushed a car out of the ditch, you will have some practical feel for this. It takes a lot of work (Fd, force over a distance) to get a vehicle from zero velocity to any velocity at all. The heavier the vehicle (m), the harder it is; the farther you have to push it (d), the harder it is. Once, the vehicle is moving, it is just as hard to stop it (as our daughter learned when her friend’s formerly stuck car rolled driverless and brakeless into the neighbor’s garage).

From the car example, we learn that the velocity of a microcosm cannot be increased or decreased instantaneously. The increase or decrease must occur over some distance and take some time. In the KE = ½mv2 equation, we get the distance by multiplying the average velocity ( ½ v) times the time it takes to reach the final velocity, vf. For instance, to accelerate a car from 0 to 60 mph in 10 seconds would take a distance of 440 feet (30 mph X 10 s or 44 ft/s X 10). That gives us the “d” in the Fd = mad equation. The mass, m, is assumed to be constant. Acceleration is the change in velocity. A change in velocity from 0 to 60 mph over 10 seconds is an acceleration of (0 + 60)/10 = 6 mph/s. That is, we increased the velocity by 6 mph for each second that we held the pedal to the metal.

Stopping a microcosm in wet clay involves the same process in reverse. Velocity must drop to zero as the microcosm transfers its motion to the wet clay over the period in which it decelerates. As shown by the math manipulation done in the KE illustration above, time cancels out and velocity appears twice. I find the KE = mad equation to be a bit more intuitive. Kinetic energy then becomes what happens to a mass as it accelerates or decelerates over distance.


god particle said...

I'm really stuck on V squared. Your explanation helps but I think you are missing the forest for the trees though... On an intuitive level one would expect that if one drove a car into a wall at twice the speed it would hit twice as hard. Not four times as hard. This counter intuitive physical impact force doesn't happen because there are tow V's in the equation when you rearrange it. But rather indicates something strange and counter-intuitive is going on. Gravity is indistinguishable from acceleration. Motion in constant gravity is constantly accelerating.

Glenn Borchardt said...

Comment 20140812 god particle

god particle:

So glad to hear from you. According to my assumptions, you were not supposed to exist. Oh well, at least you have lower case humility…

About that v squared stuff… Remember that for describing the motion of a microcosm travelling through “empty space” we use only one v in the matter-motion term for momentum:

P = mv

Doubling the velocity would double the momentum. Your intuition about hitting a wall “twice as hard” would be correct only if the “wall” was “empty space.” Once the macrocosm contains something to hit, everything changes. The velocity of the microcosm has to decrease from v to 0. During that interval, the average velocity will be: (v + 0)/2 = ½ v. We then must describe the motion of the microcosm and its rapid decrease to zero with a different matter-motion term: kinetic energy:

E = mv (½ v) = ½ mv^2

The microcosm travelling through “empty space” thus has motion we can describe either as momentum or energy. In either case, we must hold fast to the Fifth Assumption of Science, conservation (Matter and the motion of matter can be neither created nor destroyed) and the Fourth Assumption of Science, inseparability (Just as there is no motion without matter, so there is no matter without motion). In the momentum situation, it is clear that both assumptions hold as the microcosm passes from left to right through “empty space.” The momentum and kinetic energy of the microcosm remain unchanged.

When the macrocosm presents an impenetrable barrier, the motion of the microcosm stops. Whether calculated as either momentum, force, or kinetic energy, all will be reduced to zero during the collision when the velocity drops to zero. According to conservation, of course, that motion must go somewhere, perhaps as internal submicrocosmic motion of the microcosm and vibratory motion of the macrocosm. According to Newton’s Third Law of Motion, a decrease in velocity of the collider requires an equal and opposite increase in velocity of the collidee. If the microcosm was a vehicle, we would use the brakes to decrease its velocity to zero. The motion of the vehicle is transmitted to the brakes, tires, and pavement, generally appearing as the vibratory motion we call heat. Again, the second v shows up in the equation because we use it to calculate the displacement produced by the collision.

Leslie Lim said...
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