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 $t=3.0 \mathrm{~s}$. (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

Average Velocity

Average velocity is defined as the total displacement divided by the total time taken. Unlike instantaneous velocity, average velocity provides a measure of the overall motion over a specified time interval, offering a simplified overview of the change in position despite any variations in the motion between the endpoints.

Critical Points and Extremum Analysis

Critical points occur where the derivative of a function (such as velocity when considering position) is zero or undefined, indicating potential maxima, minima, or points of inflection. In kinematics, setting the derivative of the position function (i.e., the velocity) to zero helps identify when a particle reaches a maximum or minimum position along its path.

Acceleration via Differentiation

Acceleration is the rate at which the velocity of a particle changes over time, obtained by differentiating the velocity or the second derivative of the position function. It provides insight into how quickly the particle is speeding up or slowing down, and plays a key role in understanding the dynamics of the motion.

Position Function

The position function defines the location of a particle at any given time. It forms the basis for determining other kinematic quantities. In the context of motion along a single axis, the function provides a way to relate time to the positional coordinate, which can then be evaluated at specific time instances to find where the particle is located.

Kinematics

Kinematics is the branch of mechanics that describes the motion of objects without considering the causes of motion. It relies on relationships between position, velocity, and acceleration, which are often expressed as functions of time, enabling the analysis of particle motions along a line or in space.

Velocity via Differentiation

Velocity is the rate at which a particle’s position changes over time. It is found by differentiating the position function with respect to time. This concept is central in kinematics as it gives the instantaneous speed and direction of a particle, and is crucial in identifying changes in the motion such as when the particle stops or changes direction.

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