A particle moves in a straight line with the given velocity...

A particle moves in a straight line with the given velocity (in meters per second). Find the displacement and distance traveled over the time interval, and draw a motion diagram like Figure 3 (with distance and time labels).
$$
v(t)=t^{-2}-1, \quad[0.5,2]
$$

Key Concepts

Displacement

Displacement is the net change in position of a particle over a time interval, calculated by integrating the velocity function over that interval. It takes into account the direction of motion, so positive and negative contributions can cancel each other out, yielding a possibly smaller net displacement than the total distance traveled.

Distance Traveled

Distance traveled is the total length of the path taken by the particle, regardless of direction. It is computed by integrating the absolute value of the velocity function over the given time interval, ensuring that all motion, whether forward or backward, contributes positively to the total distance.

Velocity Function Analysis

The velocity function describes how fast and in which direction a particle is moving at any given time. Analyzing the velocity function includes investigating where it is positive, negative, or zero, which indicates the direction of motion and any changes in direction. This analysis is vital for correctly setting up the integration needed to compute displacement and distance traveled.

Integration

Integration in this context refers to the process of summing the continuous effect of the velocity function over time to compute displacement and distance. It is a fundamental technique in calculus that allows one to recover original quantities (like position) from rate functions (like velocity), and to compute accumulative measures such as distance traveled when using the absolute value of the velocity.

Motion Diagram

A motion diagram is a visual representation of the movement of a particle over time, typically showing discrete positions along a path at various time intervals. It helps to illustrate the particle's displacement, changes in direction, and relative speeds, serving as a useful tool to qualitatively understand the motion captured by the velocity function.

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