The graph in the st-plane of a position function s(t) of a...

The graph in the st-plane of a position function $s(t)$ of a particle moving rectilinearly is given in FIGURE 4.1.6. Complete the accompanying table by stating whether $v(t)$ and $a(t)$ are positive, negative, or zero. Give the time intervals on which the particle is slowing down and the intervals on which it is speeding up.

Key Concepts

Sign Analysis and Graph Interpretation

Sign analysis involves determining the signs (positive, negative, or zero) of the velocity and acceleration functions based on the behavior of the position graph. This analysis helps in identifying intervals where the motion changes, since the slope of the position graph corresponds to velocity, and the rate of change of the slope corresponds to acceleration.

Acceleration Function

The acceleration function, a(t), is the derivative of the velocity function, or equivalently the second derivative of the position function. It quantifies the rate at which the velocity is changing over time, with its sign—positive or negative—revealing whether the velocity is increasing or decreasing.

Speeding Up and Slowing Down Criteria

A particle is speeding up when its velocity and acceleration have the same sign; this means that the magnitude of the velocity is increasing. Conversely, the particle is slowing down when the signs of velocity and acceleration are opposite, indicating that the velocity's magnitude is decreasing.

Position Function

The position function, s(t), describes the location of a particle along a line as a function of time. Its graph provides information about where the particle is at any given time and serves as the basis for determining other motion characteristics.

Velocity Function

The velocity function, v(t), is the first derivative of the position function with respect to time. It indicates the particle’s instantaneous rate of change of position; a positive velocity means movement in one direction, a negative velocity means movement in the opposite direction, and a zero value indicates that the particle is momentarily at rest.

Please give Ace some feedback