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CHAPTER 5: MOTION AND GRAVITY (with a small amount of additional material) In this chapter, we will discuss the basic laws and concepts of “classical physics”: motion; vectors; force; Newton’s laws of motion; Newton’s law of universal gravitation; and momentum and collisions. In a more traditional physics course for science majors, a great deal of mathematics (including calculus) would be required and utilized. In this course, while a little math will be employed, it will be kept to a minimum. Not all of the material in this chapter will be covered. Rather, the focus will be on that which is necessary to address your questions. An Historical Note The first significant attempt to develop a physical theory of motion was due to Aristotle. Though his view of motion was seriously flawed, it was widely accepted for nearly 2000 years. Basically, it suggested that forces on Earth were either “natural” (e.g. “gravity”, which explains why an object falls) or “unnatural or violent” (e.g. wind, a human “push”). 2 Among the flaws in Aristotle’s view: 1. As well as a natural force of “gravity”, he proposed a natural force of “levity” (which could cause smoke to rise into the air). We know now that this is really a result of gravity. 2. Aristotle’s forces were all localized to Earth, and could not explain “universal” motion, like that observed in the heavens. It was not until the 1500s and 1600s AD that significantly different theories of motion were proposed, thanks to the work of Galileo, Newton and others, with Newton’s work being elegant, groundbreaking and still widely useful to this day. In last century, we have come to understand that Newton’s theory, while broadly valid in explaining most motions observed here on Earth, is itself flawed and limited in its scope. Specifically, while Newton’s laws continue today to provide accurate predictions of many “everyday” motions, other motions involving extremely high speeds or extremely large masses are currently best explained using Einstein’s relativity theory, while still other phenomena best fit Quantum physics. As we have pointed out before, a “theory” is just that. It can never be proved right, and may someday be proved wrong, or at least (as is the case with Newton’s laws) be shown to have only limited application. 3 DESCRIBING MOTION Motion is obviously detected as some sort of spatial change over time. That is, without the ability to measure and record length (or distance) and time, we would be unaware of motion, or at least unable to describe it in any concrete, quantitative way. So, as we will see, all of the quantities used to describe motion involve space and/or time. Fundamental Concepts of Motion Here are the basic concepts used in detecting, observing and describing motion. 1. Position (or location) Since motion must involve a change in position, this is the most basic concept of all. In order to describe the location or position of an object, we need some sort of coordinate system (e.g. Cartesian coordinates in 2- or 3-dimensions, a common street or road map, latitude/longitude on the Earth’s surface, etc.). For the purpose of this discussion, we can use a letter (e.g. x, y, z, r) to represent a body’s (or a particle’s) position. Position is obviously measured in units of length (m, cm, km, etc.) 4 2. Displacement (or change in position) Displacement is just the difference between a body’s (or particle’s) position at an initial time (or at an earlier time) and its position now (or at a later time). In mathematics, we often use the “delta” symbol to depict “change”, so for example, we could write x x f x i where xf and xi stand for the final (later) and initial (earlier) position coordinates. Like position, displacement is also measured in units of length. For example, suppose we use positive and negative numbers to represent positions east and west of North Bay, respectively. Then Mattawa and Sudbury might be located at (say) x = +55 km and x = –125 km. If you drove from Sudbury to Mattawa, your displacement would be x 55 125 180 km (i.e. 180 km east), while driving from Mattawa to Sudbury would produce a displacement of x 125 55 180 km (or 180 km west). Note that the formula automatically distinguishes east and west by assigning the appropriate sign! In other words, displacement has an associated direction attached to it. 5 3. Distance Distance is the length of any path traversed as an object moves. We often use the letter d for distance, as in the earlier formula d = ct used to describe the distance traveled by light (or radio waves) through space over time. Again, distance is measured in units of length, but, unlike displacement, distance has no associated direction. It might be tempting to think, otherwise, that distance and displacement are the same. In fact, they are if the motion takes place in a straight line and in one direction, but otherwise they are different! For example, consider a body that starts out at point P and, after traveling for some time, eventually returns to that same location. P In this case, the distance d would be the total length of the the “squiggly curve”, but the displacement x = 0! (In fact, any “closed path” that ends where it begins would have nonzero distance but zero 6 displacement.) Thus, while “distance” and “displacement” are sometimes used interchangeably in everyday conversation, they have different meanings in physics! 4. (Average or Instantaneous) Speed Speed is “how fast” a body travels, or the rate at which distance is covered. If we denote speed in general by v (we used c earlier when it referred specifically to the speed of light), then we have already seen that distance d, elapsed time t and speed v are related by d (in m/s, km/h, etc.). v t Alternatively, of course, this equation can be rearranged into the equivalent forms d d vt, t v If the speed is calculated over a “reasonable” length of time (e.g. several seconds or longer) we think of it as an average speed over that elapsed time. On the other hand if you imagine a time of a mere fraction of a second, then the speed will essentially become instantaneous. (The speedometer in a car indicates the speed at any moment in time, and is thus showing instantaneous speed.) 7 5. (Average or Instantaneous) Velocity In the same way that displacement is somewhat like a distance, velocity is somewhat like speed. In particular, velocity has an associated direction. The easiest way to incorporate this into the definition is to define velocity in terms of displacement (which has a direction), rather than in terms of speed. So, we can define velocity (also denoted by v) by velocity = displacement/time interval, or x x f xi v (in m/s, km/h, etc.) t t f ti Returning to our Mattawa-Sudbury example, if you left Mattawa at 1:00 PM and arrived in Sudbury at 3:00 PM, your average velocity would be x f xi 125 55 180 v 90 km/h (west). t f ti 2 2 As with speed, if the time interval t was imagined to be extremely small, i.e. was approaching zero, then the average velocity would become instantaneous velocity. Since velocity always has an associated direction while speed does not, we see that, in physics, “speed” and “velocity” are different, even though the two words are often used to mean the same thing in ordinary conversation. 8 In particular, as we are about to see, an object can have a changing velocity even though its speed is constant! 6. (Average or Instantaneus) Acceleration Even though we often associate acceleration solely with a change of speed, this is not really the case. Formally, acceleration is defined as the rate of change of velocity, i.e. v v f vi a (e.g. in m/s per s, or m/s2). t t f t i It is especially important to note that, since velocity always has an associated direction, then velocity can also change when the direction of motion changes (even if the speed stays the same)! Think of a car which is turning a corner (or negotiating a curve) at constant speed. As it travels around a curved path, its velocity changes direction. Therefore, it experiences a velocity change v, and so it accelerates. In exactly the same way, a body traveling in a (nearly perfect) circle at (nearly) constant speed, such as the Moon as it orbits the Earth, is actually accelerating. This may take some getting used to, if you are accustomed to associating acceleration with a change of speed rather than a change of velocity! 9 Just as for speed and velocity, an acceleration will be an average acceleration if the change in velocity occurs over a significant time interval, but becomes instantaneous acceleration if the time interval shrinks to zero. As one example, suppose you start from rest, and begin moving east until you reach a final velocity of 10 m/s (east) five seconds later. Then your average acceleration is v f vi 10 0 a 2 m/s2 (east). t f ti 5 For another example, imagine a car whose brakes are applied, slowing it from 24 m/s (east) to 6 m/s (east) during a 6-second interval. Then, its average acceleration is v f vi 6 24 18 a 3 m/s2 (west). t f ti 6 6 (Note, in this case, since the car is slowing down, that the acceleration is actually directed opposite to the car’s motion!) Shortly, we will see that the acceleration associated with an object traveling around a circle at constant speed is actually directed toward the centre of the circular path.