Suppose the graph shown in FIGURE 2.7 .18 is that of...

Suppose the graph shown in FIGURE 2.7 .18 is that of position function $s=s(t)$ of a particle moving in a straight line, where $s$ is measured in meters and $t$ in seconds. (a) Estimate the position of the particle at $t=4$ and at $t=6$. (b) Estimate the average velocity of the particle between $t=4$ and $t=6$ (c) Estimate the initial velocity of the particle-that is, its velocity at $t=0$ (d) Estimate a time at which the velocity of the particle is zero. (e) Determine an interval on which the velocity of the particle is decreasing. (f) Determine an interval on which the velocity of the particle is increasing.

Suppose the graph shown in FIGURE 2.7 .18 is that of position function $s=s(t)$ of a particle moving in a straight line, where $s$ is measured in meters and $t$ in seconds.
(a) Estimate the position of the particle at $t=4$ and at $t=6$.
(b) Estimate the average velocity of the particle between $t=4$ and $t=6$
(c) Estimate the initial velocity of the particle-that is, its velocity at $t=0$
(d) Estimate a time at which the velocity of the particle is zero.
(e) Determine an interval on which the velocity of the particle is decreasing.
(f) Determine an interval on which the velocity of the particle is increasing.

Key Concepts

Position Function

A position function describes the location of a particle along a line as a function of time. It is typically expressed as s(t) and its graph provides a visual representation of the particle's movement, with specific positions corresponding to different time values.

Average Velocity

Average velocity is defined as the change in position divided by the change in time over a specified interval. It is calculated using the formula (s(t2) - s(t1)) / (t2 - t1), and represents the overall rate of motion between two points in time, as seen from the position graph.

Instantaneous Velocity

Instantaneous velocity is the rate of change of the position function at a specific instant and is found by taking the derivative of the position function, s?(t). Graphically, it is estimated by the slope of the tangent line to the curve at the point of interest.

Tangent Line Approximation

Tangent line approximation involves drawing a line that just touches the curve at a given point. The slope of this tangent line gives an estimate of the instantaneous velocity. This technique is essential for understanding how a particle’s velocity behaves at a precise moment.

Zero Velocity (Rest Points)

Zero velocity occurs when the derivative of the position function is zero, indicating that the particle is momentarily at rest. Graphically, this is seen where the tangent to the position curve is horizontal, which might indicate a transition point in the particle’s motion.

Increasing and Decreasing Velocity

The concepts of increasing and decreasing velocity relate to how the velocity function changes over time. If the velocity is increasing, the slope of the position function (or the derivative) is becoming larger, which is often tied to positive acceleration. Conversely, if the velocity is decreasing, the slope is becoming smaller (more negative if moving in the opposite direction), corresponding to negative acceleration. This can be observed by looking at the curvature of the position graph.

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