The position of a particle moving along an x axis is given...

The position of a particle moving along an $x$ axis is given by $x=12 t^{2}-2 t^{3}$, where $x$ is in meters and $t$ is in seconds. Determine (a) the position, (b) the velocity, and (c) the acceleration of the particle at $t=3.5 \mathrm{~s}$. (d) What is the maximum positive coordinate reached by the particle and (e) at what time is it reached? (f) What is the maximum positive velocity reached by the particle and $(\mathrm{g})$ at what time is it reached? (h) What is the acceleration of the particle at the instant the particle is not moving (other than at $t=0$ )? (i) Determine the average velocity of the particle between $t=0$ and $t=3 \mathrm{~s}$.

The position of a particle moving along an $x$ axis is given by $x=12 t^{2}-2 t^{3}$, where $x$ is in meters and $t$ is in seconds. Determine (a) the position, (b) the velocity, and (c) the acceleration of the particle at $t=3.5 \mathrm{~s}$. (d) What is the maximum positive coordinate reached by the particle and (e) at what time is it reached? (f) What is the maximum positive velocity reached by the particle and $(\mathrm{g})$ at what time is it reached? (h) What is the acceleration of the particle at the instant the particle is not moving (other than at $t=0$ )?
(i) Determine the average velocity of the particle between $t=0$ and $t=3 \mathrm{~s}$.

Key Concepts

Position Function

The position function describes the location of a particle along a line as a function of time. It provides a mathematical representation of the particle’s trajectory and is the foundation for determining other kinematic quantities.

Velocity

Velocity is the rate of change of position with respect to time. It is calculated as the first derivative of the position function and indicates both the speed and the direction of the particle’s motion at any given instant.

Acceleration

Acceleration measures the rate of change of velocity with respect to time. It is obtained by taking the derivative of the velocity or the second derivative of the position function, and it indicates how quickly the speed or direction of the particle is changing.

Derivatives in Kinematics

Derivatives are essential in kinematics because they link the position, velocity, and acceleration of a particle. By differentiating the position function, one finds the velocity function, and by differentiating the velocity function, one obtains the acceleration function, allowing for a detailed analysis of motion.

Critical Points and Extrema

Critical points occur when the derivative of a function equals zero or is undefined. In kinematics, setting the derivative of the position or velocity function to zero helps locate maximum or minimum points, such as determining when a particle reaches its maximum displacement or maximum speed.

Average Velocity

Average velocity is defined as the total displacement divided by the total time interval over which the motion occurs. This concept provides a measure of the overall change in position per unit time, regardless of any variations that occur within the time interval.

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