Some physics fallacies seen on the Internet

Occasionally, I see naive examples of physics reasoning on the Internet (or signs of such reasoning behind the scenes) or, more rarely, in real life. Some interesting and partially overlapping cases that I have seen over the years:

1. The idea that body mass, in and by it self, is an advantage in the shot put.

   In reality: sometimes it is; sometimes it is not; sometimes it is an outright hindrance.

   What matters in the shot put, in terms of distance achieved, is the speed* and direction of the shot as it leaves the hand and from what height it leaves the hand.** If we assume that we have two putters of equal release height*** and with perfect technique, this reduces**** to speed. To some approximation, then, shot put is a matter of speed and the “physical” aspect boils down to getting as much speed as possible into the shot. Indeed, the overall distance reached is approximately proportional to the speed squared (cf. excursion on “high-school parabolae”).

   *For simplicity, I will speak of “speed” or a combination of “speed” and “direction”/“angle”/whatnot throughout, and ignore the term “velocity”. (In physics, speed is a scalar measure, e.g. “10 m/s”, while velocity is the combination of speed and direction, e.g. “10 m/s along the x-axis”, usually expressed as a vector.)

   **Ignoring the likely small effects of shot–air interaction.

   ***Height at which the shot is released, as opposed to how tall the putter is. The former is also influenced by e.g. arm length and release angle/direction. This height is, all other factors equal, a positive, as can be seen by applying high-school parabolae physics; however, considering the low variation in height between competitors, not that major a factor. Note that while body mass correlates with tallness, and therefore with release height, it is not the body mass, per se, which helps—it is the tallness (and length of arm, whatnot).

   ****With some other implicit assumptions, e.g. that a technique that allows a higher speed does not require a compromise in angle or height. Such factors might very well have an effect, but likely not a very large one and this item is intended to briefly show a principle—not to provide the definite work on the physics of shot put.

   Now, what matters here is how fast the putter can move his hand relative the ground, while holding the shot. The overall movement can be quite complex and includes component movements of the legs relative the ground, the upper body relative the legs, hand and arms relative the upper body, and whatnot, but it all boils down to the speed of the hand.*

   *Here, again, size can be beneficial, e.g. in that a longer arm might allow for a longer acceleration phase, which, at a fix force or power, results in a higher speed—but it is, again and beyond mere correlation, not a matter of body mass.

   In a next step, any average Joe has the natural speed to move his hand faster than even an Olympic-level shot putter—as long the shot putter is holding a shot and Joe is not. The shot has a lot of mass of its own, and hence additional inertia, which implies that the hand with a shot will be slower than the same hand when empty.* What is needed to get the maximum speed into the shot is, then, explosive strength—but, at a given level of explosive strength,** more body mass is a hindrance, because some of that strength will be wasted on moving the extra mass.

   *And, again, there might be some other factors to consider, say, that keeping technique up with while moving a shot (or a heavier shot) can be harder and require more strength.

   **And arm length and whatnot.

   All other factors (e.g. proportions of fast- and slow-twitch muscle fibers) equal, there might then be an advantage to building muscle to increase explosive strength, but not in carrying around extra fat. On the outside, the fat is a necessary evil to allow sufficient muscle growth at a given level of training; but it becomes dead weight and an unnecessary evil beyond a certain quantity.* Even muscle is not necessarily the main thing, beyond some minimum,** and muscle will likely suffer from diminishing returns; while it might make sense to train more for explosiveness than muscle mass.

   *I have no insight into what that quantity might be, but I suspect that it is comparatively little.

   **But note that this minimum might be much higher for a shot putter than for, say, a javelin thrower, due to the mass of the shot relative the javelin. I am by no means suggesting that shot putters should forget about the gym and go for a long daily run instead.

   The key to the shot put is, then, to be strong, explosive, and athletic—not to be heavy for the sake of being heavy.

   (Also see a few excursions.)

2. The idea that a lighter/heavier shot should result in a proportional change in the distance. For instance, the current men’s world record is 23.37 m, the men’s shot weighs 7.26 kg, and the women’s shot weighs 4 kg, which would give a (naive) estimate of 23.37 x 7.26 / 4 or about 42.42 m for a put of equal worth with the lighter shot.* (Among male competitors. A comparison between male and female competitors is a different story altogether.)

   *I have used Wikipedia on shot put for numbers.

   In reality, this fails on the importance of speed. The key question is what speed was achieved in the one case and what would be achieved in the other. The answer, unfortunately, is not something that can just be solved by math, as it will involve matters like physiology and individual characteristics, e.g. in that a boy of 10* might see a bigger gain/greater loss when moving from 7.26 to 4 and vice versa, than would an adult power lifter. However, chances are that the effect will be a lot smaller than in the naive calculation. This, in part, because a lower acceleration over a fix distance implies more time to accelerate, which to some degree compensates (see excursion); in part, because the overall acceleration involves more inertia than just the inertia of the shot (ditto gravitational pull). To the latter, even just the arm movement must be seen as the sum of the mass of the shot and the arm, and assuming an effective** mass of just 2 kg, we must then compare 9.26 to 6 kg (ratio 1.54), instead of 7.26 to 4 (ratio 1.82). Looking at the body as a whole,*** we might have e.g. 107.26 to 104 (ratio 1.03!). The distance to speed-squared relationship increases the difference again, but, on the balance, a smaller to much smaller difference is to be expected.****

   *Ignoring such details as what shots he is allowed to use and what shots are safe for him to use.

   **Not all mass contributes equally: that of the hand will contribute in a manner similar to the shot; that of the upper part of the upper arm much less; other parts differently yet. Whether 2 kg is realistic, and/or for whom it might be realistic, I leave unstated.

   ***Here, too, some type of effective mass should be used and the overall movement likely broken into smaller components, but I only try to illustrate the principle. The exact numbers and outcomes are secondary.

   ****More empirically, we can look at junior athletes, who often compete with multiple shot weights in the same year. Consider Konrad Bukowiecki, one of the most successful juniors. Looking at the given competition record (but note that this is not a full record, and that it might not give the whole truth), we see e.g. 2014 efforts of 23.17 m and 22.06 m with the 5-kg resp. 6-kg shot. With a scaling factor of 6/5, the longer distance likely would have been in excess of 26 m, even allowing for some day-to-day variation. Repeating this for 2015, we see efforts of 22.62 and 20.46 with the 6-kg resp. 7.26-kg shot. A scaling factor of 7.26/6 would have made the longer effort well above 24 m.

   As an aside, it is not a given that the number one with a heavier shot will be the number one with a lighter shot, and vice versa, as the putters’ ability to reach a high speed might be differently distributed for the different shots.

3. The idea that the “heaviness” of boxer’s punch is determined (solely) by the momentum and/or kinetic energy of his gloved hand.* (And, yes, I have seen several persons allegedly knowledgeable about boxing make this error.)

   *Momentum = mass x velocity (mv), kinetic energy = 1/2 mv^2.

   In reality, this is just one of the two aspects to consider.* The other is the follow-through, the continued pressure upon impact.

   *Even discounting whether just the hand is enough, or whether, as above, some type of “effective mass” for the whole arm or, even, a larger part of the body applies. It is virtually certainly the second, but, once past the arm, it becomes tricky to draw the border to my main objection, and I will not attempt a separate discussion.

   For a simple experiment, just put your fist against your cheek and jaw bones and apply increasing pressure—and note that there is neither momentum not kinetic energy involved. Now imagine that a trained boxer or other martial artist does the same at the end of a punch, while using his full shoulder, hip, whatnot strength.

   (Also note typical pieces of advice, e.g. that a boxer should “plant his feet” or “push from his feet”, and that plank-breaking karateka should aim for a point well behind the plank.)

   To make matters more complicated, different punchers can have different effects, strengths, and weaknesses, depending on how fast, heavy, whatnot they punch. (And, of course, an unexpected punch on the right spot might knock someone down where a “heavier” punch that was expected and hit in the wrong spot might not.)

4. The idea that bigger men are stronger at everything than smaller men. (At the same training level and whatnot, of course.)

   In reality, this depends on what is measured. At a naive level, we can e.g. note that someone 10 percent taller, all other factors equal, will have muscles with a 21 percent larger cross-section, implying a naive increase in strength of those 21 percent.*/**

   *1.10^2 = 1.21, which corresponds to 10 respectively 21 percent more.

   **For simplicity, I will ignore issues like the possibility of tall and short competitors differing in body proportions or ability to put on muscle.

   But let us translate this to body weight exercises. Take push-ups:

   Firstly, while muscle strength is determined by cross-sectional area, body mass is determined by volume, and that 21 percent increase in strength is bought at a cost of a 33.1 percent mass increase (1.10^3 = 1.331). This alone makes the push-up 10 percent harder.*

   *Measured strictly in a sense of strength/weight ratio. The actual practical effect can be even larger, especially when someone is approaching his maximum effort. For instance, there is no guarantee that someone who manages two reps at one weight manages even one at a 10 percent higher weight. I will ignore this complication below.

   Secondly, we have the issue of levers. Looking at his arm when bent at 90 degrees, e.g., it is obvious that the 10 percent longer arm results in a 10 percent longer lever—and a lever working against him.* This makes the push-up another 10 percent harder.

   *Generally, parts of the body tend to be levers to our disadvantage. This goes a long way to explain why straight-wrist biceps curls are considered harder than bent-wrist ones and why arm-wrestlers tend to bend their wrists.

   Thirdly, there is the issue of distance to travel. Going from nose-to-the-floor to straight-arms (or otherwise equivalent start/stop positions) that 10 percent strikes again.

   In conclusion, the taller man must exert 21 percent more force for a 10 percent longer distance. Not a good bargain. (And note, e.g., that high-level gymnasts, who are very body-exercise centric, tend to be small.)

   He fairs better with weights, but not as well as the naive might think: The first step with the increase of own weight disappears,* but the other two remain. Yes, all other factors equal, the taller man will now be 21 percent stronger, but this is reduced to 10 percent after considering length of lever (1.21/1.10 = 1.10), and he must now move the weight 10 percent longer. Chances are then, that he will be slightly better at max lifts, but take a hit in endurance.

   *Barring, in a formal competition, the risk of being put into another weight class, which could drive up the weight to lift to beat the competition. Olympic lifters, e.g., tend to be comparatively short. This allows them to spend the “weight allowance” on muscle cross-section, not muscle volume (and bone volume, whatnot). The taller man, at the same weight, is unlikely to be competitive, as he now not only loses that 21 percent advantage but is likely to have a deficit in cross-section relative his shorter competitor, while the longer levers and the longer road to move the weight remain.

Excursion on complications when reasoning:
Despite the many reservations made for even the first item above, there are undiscussed complications, which shows how tricky a complete analysis could be. Indeed, so many occurred to me that I stopped writing them down. For instance, against mass in the first item, we have the risk that someone with more mass, but moving at the same speed, will have a greater risk of fouling the put than someone with less mass, by not being able to stay inside the circle. For instance, in favor of mass, less of the speed generated by the arm might be lost when backed by a body with higher inertia than with lower inertia. (Inertia and mass are proportional to each other. The push by the arm on the shot sees the same push, but in the other direction, on the body.) As a general factor, how far the hand/shot is in front of the circle at the time of release can matter—not just how far up they are. (And note how both this and the release height is approximately independent of shot weight, which could serve to reduce the difference in result between shots of different weights.)

Excursion on mass vs. weight and the use of “mass” and “weight”:
My choice of words results from the nature of mass and weight. (Or “intended choice of words”, as I found it hard to not automatically write “weight”, and deviations might be present.) Weight is broadly speaking what the bathroom scales show when we stand on them (with variations depending on “the weight of what”) and results from the effects of gravity on (heavy) mass.*/** When it comes to e.g. lifting, weight is more relevant than mass, as it is the opposing gravitational force that must be overcome. When it comes to acceleration under a given (net) force, however, the (inertial) mass is what matters; ditto when it comes to determining the (net) force needed for a certain acceleration. Shot put lands a little between the chairs, as the horizontal velocity component depends on inertia, but the vertical depends on both inertia (mass must be accelerated) and weight (we must overcome the downwards pull of gravity). However, considering the lightness of a shot relative e.g. the weights of a power-lifting competition, it can to some approximation be viewed as a matter of inertia.

*Depending on definitions, maybe after adjusting for some other forces. It might e.g. be or not be that a lower net force (and display on those scales) through the centrifugal effects on Earth’s rotation counts as lower weight. (I have, oddly, not considered this in the past and must leave that for a later time. The more important point is, of course, the lower net force, not what carries what name.)

**The SI unit for weight is the newton (= kg x m/s^2) not the kilogram (nor the pound, pound-force, whatnot). Above, I keep to the sloppy everyday use of saying, e.g,, “100 kg” instead of (approximately) “982 N”. However, it is notable that weight will vary depending on the local gravitational field even on Earth and might be radically different on other planets—for the same mass. Far enough from gravitational sources, weight will be practically zero.

Excursion on acceleration:
If we accelerate* a given object for a given time, the resulting speed will be linear in the acceleration (v = at) and an increase/decrease of acceleration of, say, 10 percent will give a 10 percent larger/smaller speed. However, if we accelerate over a fix distance this changes: the distance is reached when the time t fulfills d = 1/2 at^2 or t = (2d/a)^(1/2), which gives us v = at = (2da)^(1/2). Now a 10 percent increase of the acceleration gives a speed slightly less than 5 percent larger, with similarly smaller increases/decreases applying for other numbers. Comparing the above shots at 7.26 and 4 kg,** the same force applied to first the one and then the other will result in an acceleration inversely proportional to weight, and (over a fix distance) the difference in speed will be a factor of (7.26/4)^(1/2), which amounts to slightly less than 35 percent, in favor of the lighter shot. Over a fix time, it would have been a factor of 7.26/4 or more than 80 percent.

*Here I assume a uniform/constant acceleration, which is not necessarily realistic and might be very incorrect for the shot put. The general principle still holds, the demonstration is much easier, and to actually calculate with the non-uniform acceleration I would need to assume a function anyway.

**Note that I do not include the hand/arm mass corrections here. This partly because I have not specified that the fix distance actually results from an arm movement/that the force comes from an arm; partly to keep the illustration simpler. However, in the case of the shot put, corresponding further corrections would be needed, and would shrink those 35 percent even further.

Excursion on high-school parabolae:
I tried to go through this, but the result, with my current markup language and its weak math abilities, was unreadable. (Take the excursion on acceleration and imagine something an order worse.) In short, however, assuming a zero initial height (and no air resistance and whatnot), we have the traditional result of a 45 degree angle for maximum distance, and this maximum distance is proportional to the squared initial speed of the shot. Introducing the height, the optimal angle is flattened, the distance is increased, and the “distance ~ squared speed” relationship is weakened. (The size of the respective effect will vary depending on the height relative the initial speed.)

Excursion on shot putters vs. sumo wrestlers:
If we look at sumo wrestlers, gridiron defensive tackles, and similar, they seem to believe that more mass (within reasonable limits) is a positive—and they might well be right. Their situation is different from a shot putter’s, who only uses his body to accelerate the shot, while they use their bodies as tools to attack and defend against similarly sized opponents. If two sumos bump into each other, the heavier will have a natural advantage. (Again, all other factors equal, including speed at impact, skill, height, whatnot.) The shot might have a 20th of the mass of a sumo wrestler, begins at a standstill, and does not fight back.

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