The equation of motion of a particle is = t^3

The equation of motion of a particle is $ = t^3 - 3t, $ where is in meters and is in seconds. Find
(a) the velocity and acceleration as functions of $ t, $
(b) the acceleration after $ 2 s, $ and
(c) the acceleration when the velocity is 0.

Key Concepts

Critical Points in Motion

Critical points are values of time where the velocity function equals zero. These points are significant because they indicate moments when the particle momentarily stops, which can also be used to analyze changes in the direction of motion.

Differentiation

Differentiation is a fundamental concept in calculus that measures the rate at which a function changes with respect to a variable. In the context of kinematics, differentiation is applied to the position function to obtain the velocity function, and a second differentiation yields the acceleration function.

Position Function

The position function describes the location of a particle as a function of time. It is the starting point for analyzing motion in physics, and its derivatives provide insights into the particle's speed and how its speed changes.

Velocity Function

The velocity function is the first derivative of the position function with respect to time. It represents the rate of change of position, indicating both the speed and direction of the particle's motion.

Acceleration Function

The acceleration function is the derivative of the velocity function (or the second derivative of the position function with respect to time). It measures how quickly the velocity of the particle is changing, providing insights into the forces acting on the particle.

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