Center-of-mass motion in one dimension is particularly easy to deal with because all the information about it can be encapsulated in two variables: xx, the position of the center of mass relative to the origin, and tt, which measures a point in time. For instance, if someone supplied you with a sufficiently detailed table of xx and tt values, you would know pretty much all there was to know about the motion of the object’s center of mass.
A point in time as opposed to duration
In ordinary speech, we use the word “time” in two different senses, which are to be distinguished in physics. It can be used, as in “a short time” or “our time here on Earth,” to mean a length or duration of time, or it can be used to indicate a clock reading, as in “I did not know what time it was,” or “now is the time.” In symbols, tt is ordinarily used to mean a point in time, while ΔtΔt signifies an interval or duration in time. The capital Greek letter delta, ΔΔ, means “the change in…,” i.e. a duration in time is the change or difference between one clock reading and another. The notation ΔtΔt does not signify the product of two numbers, ΔΔ and tt, but rather one single number, ΔtΔt. If a matinee begins at a point in time t=1t=1 o’clock and ends at t=3t=3 o’clock, the duration of the movie was the change in tt,
Δt=3 hours−1 hour=2 hours.
Δt=3 hours−1 hour=2 hours.
To avoid the use of negative numbers for ΔtΔt, we write the clock reading “after” to the left of the minus sign, and the clock reading “before” to the right of the minus sign. A more specific definition of the delta notation is therefore that delta stands for “after minus before.”
Even though our definition of the delta notation guarantees that ΔtΔt is positive, there is no reason why tt cannot be negative. If tt could not be negative, what would have happened one second before t=0?t=0? That does not mean that time “goes backward” in the sense that adults can shrink into infants and retreat into the womb. It just means that we have to pick a reference point and call it t=0t=0, and then times before that are represented by negative values of tt. An example is that a year like 2007 A.D. can be thought of as a positive tt value, while one like 370 B.C. is negative. Similarly, when you hear a countdown for a rocket launch, the phrase “t minus ten seconds” is a way of saying t=−10 st=−10 s, where t=0t=0 is the time of blastoff, and t>0t>0 refers to times after launch.
Although a point in time can be thought of as a clock reading, it is usually a good idea to avoid doing computations with expressions such as “2:35” that are combinations of hours and minutes. Times can instead be expressed entirely in terms of a single unit, such as hours. Fractions of an hour can be represented by decimals rather than minutes, and similarly if a problem is being worked in terms of minutes, decimals can be used instead of seconds.
Position as opposed to change in position
As with time, a distinction should be made between a point in space, symbolized as a coordinate xx, and a change in position, symbolized as ΔxΔx.
As with t,xt,x can be negative. If a train is moving down the tracks, not only do you have the freedom to choose any point along the tracks and call it x=0x=0, but it is also up to you to decide which side of the x=0x=0 point is positive xx and which side is negative xx.
Since we have defined the delta notation to mean “after minus before,” it is possible that ΔxΔx will be negative, unlike ΔtΔt, which is guaranteed to be positive. Suppose we are describing the motion of a train on tracks linking Tucson and Chicago. As shown in the figure, it is entirely up to you to decide which way is positive.
Note that in addition to xx and ΔxΔx, there is a third quantity we could define, which would be like an odometer reading, or actual distance traveled. If you drive 10 miles, make a U-turn, and drive back 10 miles, then your ΔxΔx is zero, but your car’s odometer reading has increased by 20 miles. However important the odometer reading is to car owners and used car dealers, it is not important in physics, and there is not even a standard name or notation for it. The change in position, ΔxΔx, is more useful because it is so much easier to calculate: to compute ΔxΔx, we only need to know the beginning and ending positions of the object, not all the information about how it got from one position to the other.
Frames of reference
The example above shows there are two arbitrary choices you have to make in order to define a position variable, xx. You have to decide where to put x=0x=0, and also which direction will be positive. This is referred to as choosing a coordinate system or choosing a frame of reference (the two terms are nearly synonymous, but the first focuses more on the actual xx variable, while the second is more of a general way of referring to one’s point of view). As long as you are consistent, any frame is equally valid. You just do not want to change coordinate systems in the middle of a calculation.
Have you ever been sitting in a train in a station when suddenly you notice the station is moving backward? Most people would describe the situation by saying that you just failed to notice that the train was moving—it only seemed like the station was moving. But this shows there is yet a third arbitrary choice that goes into choosing a coordinate system: valid frames of reference can differ from each other by moving relative to one another. It might seem strange that anyone would bother with a coordinate system that was moving relative to Earth, but for instance the frame of reference moving along with a train might be far more convenient for describing things happening inside the train.
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