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Q: How small are atoms?
A: Atoms are roughly 10-10 meter across—that’s 0.0000000001 meter. An atom has a tiny core made of protons and neutrons. This core, called the nucleus, is even tinier, typically about 10-15 meter (0.000000000000001 meter) across. It is surrounded by a kind of cloud of particles called electrons. This cloud brings the size of the atom up to about 10-10 meter. Notice that the nucleus is roughly 10,000 times smaller than the mostly empty electron cloud, so the atom is in fact mostly empty space!
Q: Why is chaos theory interesting?
A: The aspect of chaos theory that captures the imagination is the idea of unpredictability. Imagine a complicated system that changes over time, such as the weather or a river flowing down a waterfall. Unpredictability simply means that the outcome of some initial condition is practically impossible to predict, even though it might seem that one could work it out if one knew all of the mechanical laws governing the system.
For example, it is impossible to predict precisely what will happen to a stick or a boat flowing smoothly down the river and falling down the waterfall. A small change in the initial conditions can result in a completely different outcome. The system is said to be chaotic.
The final state of a system may be important though and may depend crucially on the initial conditions. For example, a small hill or valley could chaotically affect the airflow in a geographical area, which would make it hard to predict accurately whether a cloud will produce rain or a snowstorm.
Q: How would a fusion reactor differ from the nuclear reactors we currently have?
A: The nuclear reactors we have now are fission reactors. This means that they obtain their energy from nuclear reactions that split large nuclei such as uranium into smaller ones such as rubidium and cesium. There is a binding energy that holds a nucleus together. If the binding energy of the original large nucleus is greater than the sum of the binding energies of the smaller pieces, you get the difference in energy as heat that can be used in a power station to generate electricity.
A fusion reaction works the other way. It takes small nuclei like deuterium (heavy hydrogen) and fuses them together to make larger ones such as helium. If the binding energy of the two deuterium nuclei is greater than that of the final larger helium nucleus, it can be used to generate electricity.
There are two main differences between fission and fusion. The first is that the materials required for fission are rarer and more expensive to produce than those for fusion. For example, uranium has to be mined in special areas and then purified by difficult processes. By contrast, even though deuterium makes up only 0.02 percent of naturally occurring hydrogen, we have a vast supply of hydrogen in the water making up the oceans. The second difference is that the products of fission are radioactive and so need to be treated carefully, as they are dangerous to health. The products of fusion are not radioactive (although a realistic reactor will likely have some relatively small amount of radioactive product).
The problem with building fusion reactors is that a steady, controlled fusion reaction is very hard to achieve. It is still a subject of intense research. The main problem is that to achieve fusion we need to keep the nuclei we wish to fuse at extremely high temperatures and close enough for them to have a chance of fusing with one other. It is extremely difficult to find a way of holding everything together, since the nuclei naturally repel each other and the temperatures involved are high enough to melt any solid substance known. As technology improves, holding everything together will become easier, but it seems that we are a long way off from having commercial fusion reactors.
Q: What causes friction?
A: Friction is a complicated combination of many different effects. On the microscopic level most surfaces are quite rough, with many hills, valleys, and crevices. When two surfaces rub together, these features catch on one another, acting to slow down the relative motion, absorb, and redistribute some of the energy of motion, resulting in the heat you detect when you rub your hands together. This is why lubricants such as oil can reduce friction: They fill in the gaps and stop the surfaces from making as much contact with each other. Chemical bonds that can form when surfaces touch each other are another cause of friction.These can sometimes contribute to friction because energy has to be put into the bonds to break them and keep things moving.
Q: Why don’t heavy objects fall faster than light ones?
A: The following formula allows you to calculate the force exerted by the Earth on an object: F=km/r2. F is the force the Earth’s gravity exerts on the object, k is the mass of the Earth times Sir Isaac Newton’s constant, m is the mass of the object, and r is the distance from the object to the center of the Earth. The force of gravity is proportional to the mass of the object—the greater the mass, the greater the force.
Newton’s second law of motion tells us about the acceleration an object feels when a force acts on it. Acceleration is the rate at which something speeds up and so tells us how fast it falls. Newton’s second law tells us that the acceleration of an object due to a force is equal to the force exerted on the object divided by the mass of the object: a=F/m. (We’ll ignore air resistance here.) According to this equation, an object’s acceleration is proportional to the force exerted on the object—the greater the force, the greater the acceleration. But the acceleration is also inversely proportional to the mass of the object—the greater the mass, the smaller the acceleration. We saw above that a greater mass means a greater force of gravity, but that effect is balanced by the fact that a greater mass means a lesser acceleration.
The masses cancel out completely, and objects of any mass will fall at an equal rate. For objects near the Earth’s surface the rate is about 9.8 meters per second per second. This means that if you let anything fall from a standstill, after one second it will be moving at 9.8 meters per second.
Q: What are some good ways to motivate or interest my 13 year old in math?
A: There are a number of important key things to emphasize. The first is really to get across the idea that math is all around you. It is part of the world. People think of math as just algebra, etc., but it is really the language and science of patterns, and patterns are all around us. So get her (or him) to look around for examples of interesting patterns. For example, does she/he have any interest in music? There are lots of patterns in music. How musical patterns make up rhythm and melody is of great interest. Or look at a piece of fruit (segments of an orange) or a vegetable (like a piece of broccoli!), or maybe something less emotive like a pinecone. The latter two have fractal patterns in them, and also an important mathematical sequence called a Fibonnaci sequence… Also emphasize just how important it is that we can describe the world around us using mathematics. Putting humans on the moon, for example, is a remarkable example. How much detail you can put into that sort of discussion may depend upon your own math background, so you will have to pick and choose. But it might be most fun to discover these things for yourself along with your 13 year old.
Then there are some great books to read. Very many. You can read them together or independently. There is a recent book about the history and modern ideas about the idea of zero! Can you believe that this number was not in the Western number system until very recently? It was invented (or discovered) in the East a long time ago, and only came to the West via Arabic number systems a few centuries ago. This sort of stuff can be very interesting to someone who is learning things perhaps not the most exciting way in school, since it brings things alive. There is a wonderful series of books by Martin Gardner (one of them is called Mathematical Puzzles and Diversions) that I recall from my youth. They are collections of great articles from when he used to write for Scientific American. They were fantastic. If you can find those, or others like them, that would also be a great start.
Q: What is an isotope?
A: The nucleus of an atom is made of protons and neutrons. The neutrons have no charge, but the protons give the nucleus a positive charge. Electrons, which have a charge equal to but opposite that of a proton, orbit in a “cloud” that surrounds the nucleus. In an ordinary neutral atom, the number of electrons orbiting in the “cloud” is equal to the number of protons in the nucleus.
The number of electrons surrounding an atom determines the atom’s chemical properties, making an atom of carbon, for example, different from one of chlorine. But since neutrons are neutral, the number of neutrons in an atom can change without changing the chemical properties of the atom. If you have more neutrons, the atom is heavier, and if you have fewer, it is lighter. Two atoms differing only by the number of neutrons they contain are called isotopes of each other. Deuterium, or “heavy hydrogen,” is the simplest example. Deuterium is chemically the same as hydrogen, but deuterium’s nucleus contains a neutron while hydrogen’s nucleus does not. Carbon 14 is another example. Ordinary carbon has six protons and six neutrons; carbon 14 has six protons and eight neutrons.
Q: What is the speed of light?
A: Light travels at more than 299,792,456 meters per second (about 186,000 miles per second) in a vacuum—extremely fast. The Sun is 150 million km (93 million mi) away from us, and light takes only about eight minutes to travel from the Sun to Earth! This is the fastest speed possible in the universe as we know it. Light travels a bit more slowly when moving through a medium such as glass, plastic, or water.
Q: In math, what’s the difference between the mean, the median, and the mode?
A: Mean, mode, and median are terms describing some of the properties of a collection of numerical data—that is, a set or sample of numbers. The data could represent results of some set of measurements, such as a survey of the height of all the people in the same class.
The mean, median, and mode are rough measures of the “average” value, and each of them helps give an idea of this average.
The mean is the most commonly given and is the sum of all of the numbers, or data points, divided by the total number of actual data points in the set. This can be misleading, though, since the mean does not tell you how typical the number that you get really is.
The median is simply the middle value that occurs when all of the numbers are placed in order (or the mean of the middle two numbers if the number of data points in the set is even). The median often helps you decide whether the mean is really a good guide to the value of the typical number.
Finally, the mode is the number that occurs in the sample most often. This information can be useful in deciding whether the mean and median are giving you good information about the sample.
Q: When people talk about the “new math,” what do they mean?
A: Well, there’s the new math, and then there’s the new new math. Both are a reaction to a perceived need to improve mathematics education, primarily at the high school level. As with anything new, there was a lot of resistance to them. Some of the reasons are hard to argue with, while others are plain wrong. Unfortunately, a high proportion of the discussion is political, rather than focusing on what young people should learn.
New math came about in the 1960s, in the form of a change to the high school math curriculum. It was a reaction to the need to improve the competence of United States students in all the sciences. Educators correctly identified that such improvement begins at the school level.
The traditional way of learning how to add, subtract, multiply, and divide was to memorize a lot of rules and then practice them a lot until you could tackle a fixed set of tasks extremely well. So students learned to multiply together two three-digit numbers using a set of rules. These rules are based on knowing by memory the multiplication table for single-digit numbers, then mixing in some rules involving carrying over digits, adding, and shifting. The same is true for long division. You can become very good at this without knowing anything about why it works.
The new math tried to teach more about the concepts of math—addition, multiplication, and others-–by teaching how and why they work, how they are similar and different, and what is really going on when you multiply the three-digit numbers. Students thus learn about the set of objects that the numbers represent, how operations on elements of the set work, and instructions for combining members of the set to give other members of the set.
The rules of multiplying the three-digit numbers are then recognized as a particular set of rules based on the fact that the numbers written are just representations of deeper quantities using a particular number system of ten digits called “base 10.” (Of course, this is not new math at all, since all of these concepts are based on pre-20th-century math!) While this seems abstract, it does lead to productive ways of thinking because when you know how things work, you can often deduce how other things work, and you are also likely to be able to create new things as well.
The problem: A student has only a small amount of time to learn basic math skills. The less structured, concept-based approach is harder to fit to all types of students to produce the desired results in the same amount of time.
Furthermore, you can argue that the entire population doesn’t really need to be able to think about number theory. What students do need are some definite tools for adding, subtracting, multiplying, and dividing so they can go out into the world and do things like balance their checkbooks.
Eventually the debate settled down, and the proportion of “new math” to “traditional math” in the school curriculum reached a (sort of) happy medium.
However, in the 1990s U.S. students began falling behind the rest of the planet in math skills, so people are worrying about teaching methods again. Enter the new new math.
Roughly, the new new math emphasizes learning by examples so that students can figure out some of their own rules for doing certain things. In that form there is a considerable danger, since figuring things out without really understanding them can lead to finding rules that don’t apply correctly in all situations.
It is always worth it to periodically reexamine how we teach skills in any area, and to ask what it is we are trying to achieve. The traditional versus new math discussion will no doubt continue to resurface.
Q: Why is it important that scientists be open-minded?
A: Being open-minded is what science is all about. The best science operates by letting our observations about nature determine what our theories of the world should be. Theories are tested by making verifiable predictions that can then be demonstrated through experiment.
One of the finest examples of this is Galileo. In the 16th century, following Copernicus, he put forth the idea that planets and other local heavenly bodies actually revolve around the Sun, not the Earth. At the time, Earth was believed to be the center of the universe, for religious reasons. The established religious community ridiculed both Copernicus and Galileo. Galileo used a telescope, which he constructed, and showed that Venus exhibits phases as it goes around the Sun (like the Moon does as it goes around the Earth) and that moons orbit Jupiter. These are both predictions of the idea that heavenly bodies can move around objects other than the Earth, an idea contrary to the prevailing view.
Galileo also demonstrated through experiments that light and heavy objects fall at the same rate. The Aristotelian view was that this rate depended on the weight of the falling object, so there was great resistance to this idea despite the experimental evidence.
The best modern-day scientists still operate in this tradition. New data comes along as we do new experiments. This data is assimilated into current theories. At some point, if overwhelming evidence from an experiment cannot support current theory, scientists abandon the old ideas. This is the exercise of open-mindedness in a controlled and fruitful way. However, accepting an idea on flimsy evidence is also not a good way of practicing science, and this can be just as bad as refusing to have an idea challenged.
Q: On a piano, if you go from middle C up an octave to the next higher C, what happens to the frequency of the sound wave? And what are overtones?
A: If you go from middle C to an octave above, to the next higher C, the sound waves will double in frequency. Middle C’s frequency is 261.63 Hertz (1 Hz = 1 cycle per second), while that of the C an octave higher is 523.25 Hz. Doubling the frequency again will get you another C an octave higher up.
Overtones are secondary tones that accompany a fundamental tone. They are produced by secondary wave frequencies that are ratios of the primary frequency. Let us start again with middle C. If you were to listen carefully, you would hear higher Cs (with frequency doubled, then quadrupled, and so on) mixed into the sound. These pitches become much quieter the higher the frequency of the overtone.
You can hear the other multiples of the original frequency as well. So the sound contains not just the doubled frequency, but frequencies three times as high, five times as high, and so on. These frequencies produce familiar notes: Three times the frequency of middle C is actually the G in the scale an octave higher. Five times is the E above that. Six times, or double three times, produces a G another octave higher. Seven times is B$, and nine times is D. Ten times, or twice five times, is another E.
Overtones occur like this for any instrument, because natural tones never oscillate at just a pure frequency. Instruments derive their character from the mixture of overtones (the relative loudness of the overtones, or harmonics), which varies from piano to violin to voice.
We have just scratched the surface of this subject though. As you go higher in overtones, you don’t quite get the notes of the standard scale as we know it, because throughout music history pitch has been established at different tunings that vary to some extent from pure frequency relationships.
Do you recognize the first few notes of the overtones of C? Let me list them once: C-G-E-B$. The first three constitute the C major chord. Adding the next pitch gives the C7 chord, which is used a lot in blues and jazz. These chords sound natural to us, because we unconsciously hear the overtone scale everywhere in nature. It is just another consequence of simple mathematics and physics!
Q: I would like to know the speed that gravity travels. I am making the assumption that since the Sun’s gravity affects the Earth and vice versa and so on, that gravitons have to move and therefore they have a speed.
A: This is a very good question. The answer is that gravity moves with the speed of light. So if the Sun was to suddenly disappear, it would take about eight minutes for us to see the light stop coming, and for the Earth to suddenly shoot off in a straight line, since the Sun is not forcing it to go around it anymore! (Work it out…the Sun is about 150 million km away, and the speed of light is roughly 300 million meters per second!)
A “graviton” is the name given to the basic particle of gravity, just like the “photon” is the one for light. Nobody has directly detected a graviton yet, but it fits everything we know that they exist. The graviton is a massless particle, and everything massless moves with the speed of light according to Einstein’s theory of special relativity. Quantum theory says that forces can be described as being mediated by carrier particles, and in particular, a force of infinite range, like gravity or electromagnetism, is mediated by a massless particle, like a graviton or a photon. More down to earth is trying to detect a gravity wave, an analogue of a light wave. These are believed to exist, as they are direct predictions of Einstein’s theory of gravity (general relativity), and they satisfy a “wave equation” which says that they move at the speed of light. The problem with direct detection of gravitons or of gravity waves is simply that gravity is very, very weak, and we need very accurate experiments using the most clever technological innovations to detect them over the noise of the rest of the world around us. (There is a Web site about one of the fantastic gravity wave detectors, called LIGO, http://www.ligo.caltech.edu)
Q: Can you settle a question? I always thought you could not go the speed of light because the faster you go, the greater your mass would become until you reach a point where your mass is infinite and therefore you would need an infinite amount of energy to move you to light speed. My friend says that as you approach the speed of light your mass decreases and you can never go the speed of light because you can never have no mass. Which is right?
A: Well, my job here is easy. Your first answer is correct, and you explained it so well there is not much else for me to say. Here is the formula, so you can see exactly what the rate of increase is:
M=m*gamma, where 1/gamma^2=(1-v^2/c^2)
Here, “m” is the mass you would have if your speed (“v”) was zero, and “c” is the speed of light (about 300,000,000 m/s). Gamma becomes very large as v approaches c, and it becomes infinite when v=c.
Since this mass (“M”) is the inertia that inhibits you from getting acceleration (speed increase) from an applied force (say, from your rocket engines), according to F=Ma, you can see you must have greater applied force for any appreciable acceleration (“a”) the closer you get to the speed of light. At some point, any real engine would just give up!
Q: Has there been a successful fusion experiment other than a hydrogen bomb? What was the setup?
A: It depends on what you mean by successful. There are no examples of sustained fusion reactions that produced more energy than was put into them. But scientists believe it is just a matter of time—although a longer time than we once thought, since it is very difficult.
The idea is that you have to bring close together the basic reagents, such as hydrogen or deuterium (heavy hydrogen) particles, so that their nuclei can combine and release energy. This is difficult because the nuclei have the same charge and thus repel one another—so you have to put in a lot of energy to make them get close together in large quantities. This means you need to “confine” them at high energies for the reaction to happen.
There are various ways of doing this. One way is to use magnetic confinement. The type of machine built to do this is called a Tokamak reactor; it is shaped like a big hollow donut and has powerful magnetic fields, which keep everything together. In the 1990s the Joint European Torus (JET) in Oxfordshire, England, and later the Tokamak Fusion Test Reactor (TFTR) in Princeton, New Jersey, did manage to achieve fusion, but not of the sustained type needed for use as a reliable energy source.
Another form of confinement is called inertial confinement. In inertial confinement, the momentum from electromagnetic radiation is used to confine the reagents. This requires a powerful source of radiation. Scientists use lasers and other sources (such as moving charged particles) to proceed. Research of this type was pioneered in Europe, and since then vigorous programs of research have sprung up all around the world, including in the United States.
A helpful article (with links) on inertial confinement is “Fusion and the Z Pinch” by Gerold Yonas. It was published in Scientific American in August 1998.
The Web sites for the Princeton Plasma Physics Lab and the JET lab provide more information and a cool photograph of magnetic confinement.
Q: What is superconductivity?
A: Electricity is the movement of particles called electrons, usually through a conductor such as a copper wire. Electricity is one common form in which we transfer energy from one location to another: from the power station to your home, for example. The atoms of a metal conductor are arranged in a regular pattern called a lattice. In normal situations the electrons bump into imperfections in the lattice structure as they move through the conductor. These collisions cause the electrons to lose energy as they move along. This energy loss, from which all normal conductors suffer, is called resistance.
Superconductors are very different, as they have absolutely no resistance! Under certain conditions, electrons move through conductors in a very different way, resulting from something called Cooper pairing. The electrons pair up as a result of a complicated interaction with the lattice structure. They effectively become a new type of particle that moves differently from the way electrons normally move. When electrons move in this new way, their properties are very different from those of ordinary electrons. They are effectively immune from energy losses due to collisions.
This new “phase” of the conductor occurs below a certain temperature, called a critical temperature. For simple metals, this temperature is very low, close to absolute zero. For more complicated substances, called high temperature superconductors, the critical temperatures can be as high as 133 K (-140°C). Scientists are trying to make substances with even higher critical temperatures, because the properties of superconductors are very interesting and useful. In addition to carrying electric currents without resistance, superconductors also forbid magnetic fields from penetrating them. This property can be used to levitate objects, perhaps one day helping an electric train reduce friction as it moves on its tracks!
Q: Talk of exhausted fossil fuels and the need for alternative fuel sources is already rampant. One of the most enticing solutions (at least in Hollywood films) seems to be fusion. How do you do what is commonly called cold fusion at room temperature?
A: Sadly, cold fusion is science fiction. While it is true that fusion is in principle an excellent and abundant alternative source of energy, we do not yet understand how to get a commercially viable, sustained fusion reaction. Fusion works by putting together small nuclei to make larger ones, and the “binding energy” released is turned into available energy. (For more details, see an earlier answer of mine on the difference between fission and fusion.) The key problem is that the nuclei that are being joined have the same charge; thus, you need energy to put and hold them together long enough for the “nuclear reaction” of fusion to happen.
Currently, this is done by heating everything up to very high temperatures, at which the nuclei collide energetically. This “plasma” of the particles then needs to be contained in something in order to keep everything together—which is difficult because the plasma is a very energetic substance. Today it is done with a special magnetic containment chamber, which also requires energy.
So far, it has proven difficult to get this arrangement to produce more energy than that used to keep it going. One day it will likely succeed, and then perhaps we will have an abundant, cheap source of energy. Until then we must wait for fusion scientists and engineers to figure out the best ways of getting it to work. It would be nice to suddenly discover how to do cold fusion at room temperature, but science and real progress rarely work that way.
Q: In a rotating wheel the linear velocities of different particles of the wheel point in all different directions. The only unique direction in space associated with the rotation is along the axis of rotation, perpendicular to the actual motion. Could you explain this mathematically?
A: Actually, there are two unique directions. Along the axis of rotation one way, and along the axis pointing the other way. There are thus two vectors (“axial” vectors, as they are called, technically speaking). You have already given the rough idea of why it has to be perpendicular to the plane. If it were not exactly perpendicular to the plane in which the rotation was happening, then it would pick out a preferred direction in which part of the wheel is moving at any instant. So perpendicular to the wheel is intuitively the best way to define the rotation. A vector has direction and size (magnitude). The vector representing a rotation has this, since the two directions along the axis and the other way represent the two choices of counterclockwise and clockwise, for the rotation. The size of the vector represents how fast the wheel is spinning.
The physical reason for having such a vector is because it represents part of an important quantity called the “angular momentum.” This is an important quantity since it is often a “conserved” quantity in a particular motion, which means that it is preserved throughout the motion. This is not an abstract concept. Linear momentum is often conserved, and you see it when an object collides with another object and makes it speed up. The original object then is not going as fast as before, since it gave the other object some of its linear momentum. You are familiar with the rule that the momentum is proportional to the speed of motion times the mass of the object. Something roughly similar happens for angular momentum. Well, you’ve probably seen ice skaters speed themselves up in a spin by pulling their arms (which are perpendicular to the spin axis) closer into their bodies making themselves a more compact object in that perpendicular plane. What is going on is that they have increased their spin speed by reducing something called their “moment of inertia.” In simple situations, angular momentum is angular velocity (that vector we talked about) times the moment of inertia, which is roughly a measure of a combination of the mass of the object and–crucially–how extended it is in that perpendicular plane. The more spread out, the more angular momentum. So when the skater’s arms are drawn in, the moment of inertia decreases, and since the angular momentum must stay the same, the rate of spin (angular velocity) must increase.
So the vector nature of rotation is in fact quite important in building up such concepts in order to describe the world, although you should be suspicious, since I have not mentioned that the direction can be conserved as well, which should be the case if it is a vector conserved quantity. Well, there are two obvious examples. Ever wondered why the Earth and all the planets all orbit the Sun in the same plane, year in, year out, and never go around in a different plane? Conservation of angular momentum. The direction of the vector is perpendicular to that plane and it just stays put. Ever wonder why it is harder to fall off a bike once it is actually rolling rather than standing still? This is also because of the vector nature of angular momentum, but it would take a bit longer to explain.
Q: Why does water expand when it freezes?
A: A water molecule is made of two hydrogen atoms bound to an oxygen atom. Liquid water consists of a random arrangement of these molecules moving around. Ice, on the other hand, is made of water in a crystal state, just like the snow crystals you might see in cold weather. At freezing temperature, 0°C (32°F), the molecules prefer to form ice, rather than stay liquid. A crystal is a more orderly shape, appropriate to the fact that at low temperatures the molecules move around a lot less. A crystal is made of the same molecules but arranged in a regular repeating pattern. The unusual thing about water is that the pattern of molecules that makes ice takes up more room than the same molecules in liquid form near that temperature. So when you freeze water, it expands.
This fact is why you should not freeze water in a closed, rigid container such as a glass bottle. The container will likely break as the water freezes and expands, sometimes quite spectacularly and dangerously.
Q: What is calculus, and what practical uses does it have?
A: Actually, “calculus” can mean any particular means of reasoning or figuring things out. That can include anything from counting on your fingers to long division.
What people usually refer to as calculus is more accurately termed infinitesimal calculus, or differential or integral calculus. This type of reasoning is used to manipulate certain mathematical objects, known as functions, that depend on one or more quantities, known as variables.
A familiar example is the case of a moving object. How do you know it is moving? Well, as time goes by, its distance from some reference point changes. We can call the distance x, for short, and the time t. The fact that x depends on t for a moving object can be stated in mathspeak as “x is a function of t.” We often write x(t) to denote this.
Now just how fast is the object moving? We’d measure this by finding out how far it has moved after some fixed interval of time, like one second, or one hour. So if it moves 1 m in 1 second, we say it is moving at a speed of 1 m/s, or if it moves 10 mi in 1 hour, we say it is moving at a speed of 10 mph. One object is moving faster than another if it has gone farther in the same amount of time.
The problem, as seen in the last example, is that this might not be accurate enough. During the course of the hour, the object (such as a car on a journey) might have been moving at different speeds from time to time—-to stop at the traffic lights or to speed up to pass another car, for example. A better representation of the speed might be found by measuring the changes in distance over a smaller time interval, perhaps 1 second. The length of the interval depends on how accurate we need to be.
Calculus works with these quantities. We say that the speed is the “rate of change” of distance with time, or that the speed is the “derivative” of x with respect to t, denoted s = dx/dt. This is arrived at as follows: If you imagine taking the smallest time interval, denoted dt, over which you want to measure the distance change, dx, the result you would get for the speed is dx/dt. This is the best way of expressing the speed at any instant. These tiny or “infinitesimal” intervals are the sort of quantities that calculus works with to describe the properties of functions that often describe some physical situation very accurately.
Now we can ask about the change in the speed itself. A change in speed is an acceleration or a deceleration. How rapidly that change occurs is a useful thing to know. The quantity representing the instantaneous change of speed, a = ds/dt, is also a calculus quantity. It is the first derivative of the speed, or alternatively, the second derivative of the distance.