Rectilinear Motion In Exercises 65-68 , consider a particle...

Rectilinear Motion In Exercises $65-68$ , consider a particle moving along the $x$ -axis, where $x(t)$ is the position of the particle at time $t, x^{\prime}(t)$ is its velocity, and $x^{\prime \prime}(t)$ is its acceleration. $x(t)=t^{3}-6 t^{2}+9 t-2, \quad 0 \leq t \leq 5$ (a) Find the velocity and acceleration of the particle. (b) Find the open $t$ -intervals on which the particle is moving to the right. (c) Find the velocity of the particle when the acceleration is $0 .$

Rectilinear Motion In Exercises $65-68$ , consider a particle moving along the $x$ -axis, where $x(t)$ is the position of the particle at time $t, x^{\prime}(t)$ is its velocity, and $x^{\prime \prime}(t)$ is its acceleration.

$x(t)=t^{3}-6 t^{2}+9 t-2, \quad 0 \leq t \leq 5$

(a) Find the velocity and acceleration of the particle.
(b) Find the open $t$ -intervals on which the particle is moving to the right.
(c) Find the velocity of the particle when the acceleration is $0 .$

Key Concepts

Solving for Critical Points in Motion

Critical points in motion occur where velocity or acceleration equals zero. Setting the acceleration to zero can help identify moments when the particle's speed is neither increasing nor decreasing, while solving for when velocity is zero indicates potential changes in the direction of motion. These critical points are crucial in the analysis of the overall behavior of the particle.

Position, Velocity, and Acceleration

In rectilinear motion, the position function describes the location of a particle along a line as a function of time. The first derivative of this function, known as velocity, represents the rate of change of the position; it indicates both how fast and in which direction the particle is moving. The second derivative, called acceleration, shows the rate at which the velocity is changing over time.

Derivatives as Rates of Change

Derivatives are fundamental in analyzing motion because they provide a systematic way to relate the position, velocity, and acceleration of a particle. The process of differentiation yields the instantaneous rates of change, making it possible to understand how a particle's motion evolves at any given moment.

Analyzing Motion Direction Using the Sign of Velocity

Determining the direction of a particle's movement involves examining the sign of its velocity. A positive velocity indicates motion in one direction (such as to the right), while a negative velocity suggests motion in the opposite direction (such as to the left). This concept is essential for identifying intervals where the particle is moving forward or backward.

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