A position-time graph for a particle moving along the x axis...

A position-time graph for a particle moving along the $x$ axis is shown in Figure $P 2.7$. (a) Find the average velocity in the time interval $t=1.50 \mathrm{s}$ to $t=4.00 \mathrm{s}$. (b) Determine the instantaneous velocity at $t=2.00 \mathrm{s}$ by measuring the slope of the tangent line shown in the graph. (c) At what value of $t$ is the velocity zero?

Key Concepts

Tangent Line

A tangent line to a curve at a given point is a straight line that touches the curve at that point and has the same instantaneous slope as the curve. In the context of kinematics, drawing a tangent line at a specific point on a position-time graph allows one to determine the instantaneous velocity by measuring its slope.

Instantaneous Velocity

Instantaneous velocity refers to the velocity of an object at a specific moment in time. Mathematically, it is the derivative of the position function with respect to time, and graphically, it is represented by the slope of the tangent line to the position-time curve at that particular point.

Zero Velocity Condition

The zero velocity condition occurs when the instantaneous velocity of a particle is zero. On a position-time graph, this is observed as a point where the tangent line is horizontal. Identifying when the instantaneous velocity is zero can provide insights into changes in the direction of motion or points of rest during the motion.

Position-Time Graph

A position-time graph visually represents the motion of a particle by plotting its position along a particular axis versus time. It is a fundamental tool in kinematics, allowing one to analyze the motion characteristics of an object by examining how its position changes over time. The slope of the graph at any segment indicates the velocity of the particle during that time period.

Average Velocity

Average velocity is defined as the total displacement divided by the total time taken over a specific interval. It provides a measure of the overall change in position regardless of the motion details within that interval. In graphical analysis, average velocity corresponds to the slope of the line connecting the start and end points of the interval on the position-time graph.

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