Chapter 1 – Excerpt 2
The next excerpt from Chapter 1 discusses velocity, acceleration and force. It contains algebra and calculus (which you may skip), a photo of one of my cats engaged in their other favorite activity (which you may dwell on to make up for skipping the math), and an exposé of the real source of Newton’s discovery of gravity under the apple tree.
As you can tell, cats are allowed on the table at our house. It would be too hard to explain to them that the tabletop is off-limits to cats when they see so many miniature statues of themselves there. Besides, it’s a fertile source of material for scientific investigation.
Velocity is a measure of changes in position over time; to be precise, it’s the rate of change of
As a mnemonic device, since velocity is the canonical example of a vector, and speed is the
Q. Suppose your person rushes you to a veterinarian at 60 miles per hour (mph) and then drives home
A. Your average speed is 40 mph, since
However, the question was about average velocity, not average speed. Although the former is a vector, it is much easier to calculate. Since you return to where you started from, your net displacement is zero; hence your average velocity is zero. In equations, vavg = ∆x/∆t and ∆x = 0. So the answer is zero, in any units.
Q. This exercise is taken verbatim from a college-level calculus-based physics textbook5. The figure to the right is a plot of the speed of a cat versus time. How far did the cat travel during the third second of its journey? What were its maximum and minimum speeds? When, if ever, was its speed nonzero and constant? Approximately, what was its instantaneous speed at each of the following times: 0, 1.0s, 2.0s, 4.5s, 6.0s, and 7.0s? During what time intervals was its speed increasing? When was its speed decreasing?
Q. Suppose you are chasing a mouse who is traveling at 3 m/s and is initially 20 meters away from you. Your motion is described by x = t2 + 4t. What is your instantaneous speed at the moment you capture the mouse?
So it takes 5 seconds for you to pounce on the mouse. Now we obtain the velocity function by differentiating the position function: v(t) = dx/dt = d/dt(t2 + 4t) = 2t + 4. So at time t=5, the speed is v(5)= 2·5+4 = 14, and hence the answer is 14 m/s.
Acceleration is a measure of changes in velocity over time; precisely, it’s the rate of change of velocity over time. Notice that while the direction of velocity is always the same as the direction of motion, the direction of acceleration may not be. It is only the same direction when the object is speeding up; it’s the opposite direction when the object is slowing down. Colloquially, negative acceleration is called deceleration.
Q. Suppose you are chasing a mouse who is traveling at 3 m/s and is initially 20 meters away from you. You are going faster than the mouse but decelerating, in such a way that every second the distance between yourself and the mouse is halved. Thus, at time t = 1 second, the mouse is 10 meters ahead of you; at t = 2 seconds, you’re only 5 meters apart; and so on. Will you ever reach the mouse and, if so, when?
A. This exercise is an example of Zeno’s Paradox. You will only reach the mouse in the limit t → ∞. However, as an engineer would say, when you are within a distance comparable to the width of an atom, you’re certainly "close enough for practical purposes". It is precisely because of such issues involving limits that Newton and Spitface developed the calculus, which allowed us to solve the previous exercise.
Although those three quantities – position, its time derivative velocity, and its second time derivative acceleration – suffice for most applications, an infinitely curious cat may wonder, what about the rest of the infinitely many higher derivatives of position? Actually, the third time derivative of position, i.e. the derivative of acceleration, does have a standardized, albeit little-known, name: it’s called "jerk" (except in England, where it’s called "jolt"). Higher order derivatives are not used often enough to have standardized names, but the fourth time derivative of position was called "jounce" when it was used in the development of the Hubble Space Telescope; and "snap", "crackle" and "pop" have also been proposed as names for the fourth, fifth and sixth derivatives, respectively6. See Figure 1-3.
Force, Mass and Gravity
A force is an action that alters, or tends to alter, a body’s state of motion. We say “tends to” because some forces aren’t strong enough to overcome counteractive forces. For example, if you push on a locked door, it probably won’t affect the door’s motion, but you’re still exerting a force on the door. Informally, if you push or pull on something, you are exerting a force on it.
A force of nature is one that exists in the natural world without any action on the part of a human or feline or machine, etc. There are four fundamental forces of nature. The first of the four to be discovered, and the one that most of us are still most aware of, is the force of gravity. (The other familiar one is electromagnetism, and the other two occur on the subatomic level. They will be discussed in chapters 3, 5 and 8.)
The discovery of gravity has been attributed to Isaac Newton, in the apocryphal falling apple story, but the credit is actually due to his cat, Spitface. As the story goes, Newton was sitting under an apple tree when an apple fell on his head, and this incident caused him to realize that there was a force pulling the apple to the earth, namely the gravitational attraction between the earth and the apple. However, when analyzing this incident, most people have failed to ask why Isaac was sitting under the apple tree in the first place that fateful day. The answer is that he was very tired and wanted to rest, but when he went to his usual resting place, his favorite overstuffed chair in the living room, Spitface was curled up on it. Had the cat foolishly chosen to sleep in a less comfortable and less safe spot, such as, for example, under the apple tree, Isaac would have rested in his chair that day, as usual, remaining blissfully unaware of the force of gravity.
The SI unit of force is called a Newton (abbreviated N). Perhaps the historical oversight is not so unfortunate after all: not only would calling it a Spitface sound ludicrous, but abbreviating it as ‘S’ would lead to confusion with the abbreviation ‘s’ for seconds.
5 – Exercises 23 & 40 on pages 63-64 in Physics: Calculus by Eugene Hecht. <return-to-text>
6 – math.ucr.edu/home/baez/physics/General/jerk.html<return-to-text>
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