Topic 1-6: Positions in one-dimensional space, speed and acceleration

To specify a position in a one-dimensional space that is either a straight line or a curve, a reference point called the "origin" must be set. All the points in the one-dimensional space can then be described by the distance from the point to the origin; the distance can be either positive or negative.

The average speed within a time interval Δt is defined as the change of distance Δx divided by Δt. If the distance between a point-like object and the origin at time t is denoted as x(t), then the average speed <v> in the time interval of t to tt is defined as

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As Δt0, the speed at t, v(t), can be defined as
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The speed v(t) can be either positive or negative. If x(t)0, and v(t)>0, then the point is moving away from the origin; if x(t)0, and v(t)<0, then the point is moving toward the origin. If x(t)0, v(t)>0 means the point is moving toward the origin, and v(t)<0 means that the point is moving away from the origin.

The average acceleration in a time interval is defined as the change of the speed divided by the time interval. By taking the time interval to zero, we can define the acceleration at time t as

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If the one dimensional space under consideration is a straight line, then the magnitude of the acceleration defined here equals the magnitude of an acceleration vector that will be defined later. However, if the one dimensional space under consideration is a curve, then the acceleration defined here has nothing to do with the acceleration vector. In that sense the acceleration defined here is better called the "scalar acceleration" in order to distinguish it from the acceleration vector. But the word "scalar acceleration" is not widely used in physics textbooks, so we just use "acceleration" in place of "scalar acceleration."