In discussing my measurement of the freefall of Building 7 with one (reluctant) physics professor, the professor at one point protested, “I’ll take your word for it, for the sake of argument, but I have no way to confirm your claims for myself.”
That comment astounded me because I see the motion of the building as such a straightforward measurement. But upon reflection I recalled all the time and energy and collaboration that went into gathering all the necessary materials and information for the measurement: a suitable video, an appropriate software tool to make measurements on the video, calibration data, knowledge of video formats, etc. I realized that even with the requisite knowledge base and skill set, it would take a very motivated person to reproduce my measurements. I therefore collected together all the bits and pieces that went into my measurement and present them here as a kit that can be used by a physics teacher or handed to a motivated physics student as a lab assignment or project or science fair entry. The skills that will be learned in the process will serve a physics student well for other projects that might involve using video as a measuring tool.
All the tools and material needed are free and downloadable here:
- Lab Instructions,
- the “Kit” with the necessary materials, (*Updated 4/9/2018)
- a link to the excellent measurement tool Tracker
- a link to a Video Tutorial on Using Tracker (search for other tutorials as well)
There is nothing particularly difficult in this measurement, but some familiarity with the concepts is necessary to understand the significance of the results.
Since not everyone has taken highschool physics, I have now gone one step further. I taught physics and math for over 35 years until my retirement, and since that time I have been recording the entire highschool math curriculum for self-teaching students, whether homeschoolers or adults returning to “fill in the gaps” in their background. Over the last couple of years I have added a physics course to the list. The complete course is available here, along with math courses from Algebra 1 through Calculus. However to meet the needs of the physics-interested public who would like to better understand the events of 9/11 I have packaged a 5-chapter subset of that course and am making it available as “Physics for 9/11,” for download at no cost. It can be downloaded here. This subset of the curriculum covers measurement, motion, forces, energy, and momentum. (If you really get into it you might want to take the whole course, but this part is free.)
As I explained to one professor, what I have done is a straightforward measurement of a publicly observable event in the public sphere that is sufficiently anomalous to arouse anyone’s curiosity. Observing and measuring what is before your eyes is not political and should not be seen as controversial. This should be legitimate fare for students at any school or college, public or private. Deciding to not observe and/or not measure something and/or to criticize someone for making a measurement because of the feared political ramifications of the result, is in fact a political statement.
[I would be very interested to hear from any teachers or students using these resources for labs, projects, science fairs, etc. — David Chandler]
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Intro to Kinematics
The goal here is to present about a semester of introductory physics in one web page, an impossible task if there ever was one, so if it is not entirely successful that’s why. If you’ve had a little physics it may help put the various ideas in context for you.
Kinematics is the study of motion. Motion involves the basic concepts of position and time and specifying position as a function of time. Putting dots on the frames of a video with Tracker or Physics Toolkit, or similar software is an exercise in kinematics.
The rate of change of position with respect to time (how fast something moves) is called velocity, which is very close to the idea of speed. The difference is that velocity is a vector, which means you have to keep track of direction as well as the speed. Vectors are represented as arrows. The length of the arrow represents the speed. A velocity vector changes if either the direction or the speed changes.
The rate of change of velocity is called acceleration. It’s how fast the velocity changes. For this “micro-course” we restrict our attention to one dimension (straight line motion), so we don’t have to worry about the distinction between speed and velocity. If you increase your speed from 0 to 60 mi/hr in 10 seconds, you are gaining speed (accelerating) at a rate of 6 mi per hr per sec. Note that there are two different units of distance mixed together there. This is equivalent to 8.8 (ft per sec) per sec. The unit of acceleration (ft/sec/sec) is abbreviated ft/sec^2 (where the ^ indicates an exponent in klutzy typewriter notation. Read it as ft per sec squared). In metric units the unit of acceleration is m/s^2. It’s how much the speed (measured in m/s) changes each second.
Bottom line: Kinematics involves the quantities: time and position, and a sequence of quantities derived from these (i.e. derivatives of position with respect to time): velocity, acceleration, jerk (which is the rate of change of acceleration with respect to time), etc. on up the line. For simple problems, such as cruising down the highway, velocity might be constant with all the higher derivatives equal to zero. For slightly more complex problems acceleration might be constant, with the higher derivatives equal to zero. This is the situation when a rock falls under the influence of gravity near the surface of the earth.
Gravity causes falling objects to accelerate (speed up as they fall). As long as air resistance is not significant, all freely falling objects accelerate at the same rate: 32 ft/s^2 or in metric units 9.8 m/s^2. (This varies slightly depending on how far you are from the center of the earth. It’s slightly less on a high mountain. It’s slightly less at the equator because the earth bulges at the equator so you’re farther from the center.) In New York City the acceleration of gravity is 9.802 m/s^2. A falling rock would increase it’s speed by 9.802 m/s each second.
Dynamics: Newton’s Laws of Motion
Newton’s Laws of Motion add two more quantities to the picture: Force and Mass. A net force causes acceleration, and mass is the tendency of an object to resist acceleration. Here are the laws:
Newton’s First Law: The Law of Inertia
In the absence of a net force (i.e. if all forces acting on an object balance out to zero), an object at rest will tend to remain at rest, and an object in motion will tend to remain in motion at a constant speed in a straight line (i.e. constant velocity).
The first part of this is easy to accept, but the second part is counter-intuitive for most people until they educate their intuition. Time for a quiz.
Q: You’re driving down a long straight road at a constant 60 mi/hr. There are forces opposing your motion (friction and air resistance) and there is the force pushing your car forward. Which must be larger?
A: Most people who haven’t fully absorbed Newton’s First Law will say the forward force must be greater, because deep down in their souls they believe it takes some force just to keep moving. What Newton’s First Law spells out, however, is that it takes zero force to keep moving in a straight line at a constant speed. Any excess force in the forward direction would make you speed up, and any excess force in the backward direction would make you slow down. What you’re doing when you keep your foot on the accelerator pedal to go a constant speed is balancing the forward and backward forces. It took an excess forward force to get you up to speed, but it takes zero net force to maintain your speed.
One of the deep implications of Newton’s First Law is that you can’t tell if you’re moving or not. That’s why you can function normally while flying at cruising altitude. It feels like you are at rest. In fact as the earth spins, people near the equator are traveling about 1000 mi/hr. We are all traveling at about 66,000 mi/hr as we orbit the sun. Yet we feel ourselves to be at rest.
Newton’s Second Law: F = ma
Maintaining your position requires no force. You can’t tell if you’re moving or not, so constant velocity requires no force. In the chain of derivatives of position with respect to time, Force and Mass come into play at the level of acceleration.
If an object experiences a net force, it will accelerate. The acceleration will be in the direction of the net force, proportional to the net force, and inversely proportional to the object’s mass.
As worded here, we could say a = F/m. This is usually turned around to read F = ma. If you attach the same outboard motor to a row boat or a battle ship, we would see that equal forces do not cause equal accelerations. The battle ship resists acceleration due to its larger mass. This tendency to resist acceleration is called inertia. Mass can be thought of as a measure of an object’s inertia. Newton’s First Law is called the Law of Inertia because objects will not accelerate if there is zero force acting on them.
If you hold a rock in the air, gravity and your hand provide opposing forces that balance out, so there is no net force and therefore no acceleration. If you let go, the force of gravity becomes an unbalanced force, so the rock accelerates downward. If you drop a rock with twice the mass it has twice the inertia, but the force is also twice as great, so the acceleration is the same. Therefore all objects moving under the influence of gravity alone accelerate downward with equal accelerations. Therefore we can talk about THE acceleration of gravity. All objects in freefall (bowling balls, buildings, etc.) will accelerate at THE acceleration of gravity (for that spot on the earth). If they encounter resistance, however (due to air resistance, structural supports underneath them, etc.) the upward resisting force combines with gravity to produce a smaller net force, resulting in a smaller acceleration. If the upward force is actually greater than the weight of the falling object, the net force will be upward, so the acceleration will be upward. An upward acceleration for an object moving downward results in a slowing of the downward motion.
Newton’s Third Law: The Law of Action and Reaction
All forces are actually interactions between pairs of objects. The forces the two objects exert on each other will always be equal and opposite.
Consider standing on a bathroom scale. You push down on the scale, which flexes a spring and turns a dial to indicate your weight. The scale provides an upward force on you to balance the downward force of gravity acting on you, so you remain at rest. Time for another quiz:
Q: You jump out of a plane. Gravity acts downward on you. What is the “reaction force” of the “action-reaction” pair?
A: The two things that are acting on each other are you and the earth. Gravity mysteriously seems to reach out across empty space and pull on things. Actually, however you and the earth are pulling on each other, with equal and opposite forces. If the earth pulls downward on you with 150 lb. of force, then you are pulling upward on the earth with 150 lb. of force. The reason it is you who does most of the moving is because of the earth’s far greater inertia. (We’re back to the outboard motor attached to a battle ship: more mass means less acceleration for a given force. )
Newton’s laws of motion (all three of them) hold very precisely except for situations of extreme speed (close to the speed of light) or extreme gravity. These are situations where the laws of relativity are needed. For non-relativistic situations, such as falling buildings, Newton’s laws (all three of them, arguments of some “debunkers” notwithstanding) work very precisely. They are universal laws.
Restating Newton’s Laws: Energy and Momentum
All questions of motion can theoretically be answered in terms of Force and Mass by using Newton’s three laws of motion. However, it is often more convenient to reformulate dynamics in terms of “conserved quantities.” When you push on an object and cause it to move, you can describe what you’re doing as “acting on” it (exerting a force on it). But you could also think of this as “giving something to it.” The idea of “giving something to it” implies that what the object gained, the other object lost. Something is preserved (conserved) in the transaction.
Work and Energy
Work is defined as a force times the distance moved in the same direction during the interaction. If you shoot an arrow, the bow string exerts a force on the arrow through some distance. By so doing, you have changed the “state” of the arrow. We say we have given it kinetic energy. If you lift a bowling ball behind you, you are exerting an upward force through some distance, so you have done work on it. This work changes the state of the ball in a different way. We have lifted it higher in a gravitational field. If we quantify the amount of lift for a given mass in terms of the amount of work done, we come up with an expression we call potential energy. When the ball swings downward and rolls down the bowling alley the potential energy has been converted to kinetic energy. The work you did takes energy out of your body (that came from the last meal you ate), converts it into potential energy, which then is transformed into kinetic energy. Energy is a kind of bookkeeping system that allows us to talk about the work we do, the height of an object in a gravitational field, and the resulting motion all in common terms. There are many other forms of energy. Frictional forces feed into random molecular motions which we experience as thermal energy. As moving objects “naturally” come to rest, all the energy has been fed into the big sink-hole of thermal energy. A ball of clay thrown across the room has kinetic energy. All of the molecules share the same motion. This kind of energy can do work (such as applying a force through a distance to break a window). If the lump of clay hits the wall and “stops,” the molecules are still moving, but randomly. The randomization of the motions has made the kinetic energy of the molecules less accessible to do work, so we say the entropy of the system has increased. This is getting into thermodynamics.
Impulse and Momentum and its Relation to Kinetic Energy
We defined work as a force times the distance through which the force was applied to an object, and the energy as what an object gains (in one form or another) when work is done on it. If instead we multiply the net force by the time the force is applied, the force times the time is called impulse, and the effect of an impulse is to change the momentum of the object. Momentum is the name for the quantity mass times velocity. If a mass with momentum collides with a second mass, it can transfer some of its momentum to the second mass, but the total momentum of the two masses after the collision remains the same as before. If two objects are both moving when they collide, the total momentum of the two bodies after the collision is equal to the total momentum before the collision.
One major difference between momentum and kinetic energy is if you have two equal mass cars moving at equal speeds but in the opposite direction, the total momentum adds to zero, because momentum is directional. One of the velocities is negative. If they hit exactly head on they will come to rest. In the case of kinetic energy, however, the velocity gets squared, so the kinetic energy of the two masses both add as positive numbers. All that kinetic energy has to go somewhere before things can come to rest. They get rid of their excess energy by “doing work” on their environment: tearing things up, smashing things, and throwing things around. So it is the kinetic energy, rather than the momentum, that tells how destructive the collision will be.
Another difference between kinetic energy and momentum is the relative importance of the mass and the velocity. Not to get too mathematical here, but it is important to recognize that the momentum of a moving mass is given by the mass times the velocity. Thus the momentum is directly proportional to the object’s velocity. Doubling the velocity doubles the momentum. However the kinetic energy is given by formula, one half times mass times the velocity squared. The important thing to notice here is that its kinetic energy is proportional to the square of the velocity. Thus, for a given mass, doubling the velocity doubles its momentum, resulting in doubling its ability to knock things down. But doubling the velocity quadruples the kinetic energy. All of that energy must be dissipated before everything can come to rest, which means it is able to do a lot of damage.
An important example of the effect of high kinetic energy on 9/11 is the effect of the high speed of the planes that hit the buildings. All of the planes were moving well over 500 mi/hr at impact. Because the velocity gets squaring in computing the kinetic energy, the damage done by a 500 mi/hr plane compared with a 100 mi/hr plane, is not just 5 times as great. It is 2500 times as great. All of that kinetic energy has to go somewhere before all the pieces come to rest, so it results in 2500 times as much damage. This is why people’s intuition for the amount of damage done in a typical plane crash is not a good guide. Most plane crashes occur near take off or landing where the plane is trying to minimize its speed. This is the source of people’s expectation of seeing “plane parts” like wings, the tail, and the fuselage lying around. The reality is, however, that all of the 9/11 planes were much more thoroughly “shredded” by the impacts, as were the walls they impacted. “Seat of the pants” estimates need to give way to actually doing the physics when the situation is far from everyday experience. A little bit of math goes a long way toward a deeper understanding. Learning physics is learning to see the world with different eyes.