01 / CHAPTER 01
Position, time, and reference frames
Choose coordinates and describe an event without smuggling in a preferred observer.
ANCHOR EQUATION
Position as a function of time
In symbols: x(t) = x₀ + v₀t + ½at²
A model for one-dimensional motion when acceleration is constant.
An origin is a physical choice
Position x is not an intrinsic property of an object; it is the signed coordinate assigned by an observer. Time t labels events, while a reference frame fixes the ruler, clock, and positive direction used to compare them. The same event can have different x values in different frames without any contradiction.
Turn a verbal motion into a model
Choose an origin x = 0 and let the object’s position be x(t). If it is at x₀ when t = 0 and moves at constant velocity v₀, equal time intervals produce equal displacements: Δx = v₀Δt. Taking the initial time as zero gives x(t) − x₀ = v₀t, so x(t) = x₀ + v₀t. The equation predicts a straight line on an x–t graph; its slope is v₀ and its intercept is x₀.
A cyclist starts 12 m east of a mark and rides west at 3.0 m/s. With east positive, x₀ = 12 m and v₀ = −3.0 m/s. After 5.0 s, x = 12 − 15 = −3 m: the cyclist has crossed the origin by 3 m, not traveled a negative distance.
When the frame becomes part of the physics
This model assumes one-dimensional motion and constant velocity. If the track bends, one coordinate is insufficient; if the speed changes, the straight-line x(t) model fails. A shifted origin changes x₀ but not the measured displacement between two events.
Work it: Draw two number lines with different origins. For x₀ = 4 m and v₀ = 2 m/s, predict x at 0, 1, and 3 s, then explain why the slope is unchanged if the origin is moved 10 m.