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

The position of a particle moving along the $x$ axis is given in meters by $x=9.75 \mathrm{~m}+\left(1.5 \mathrm{~m} / \mathrm{s}^{3}\right) t^{3}$ where $t$ is in seconds. Calculate (a) the average velocity during the time interval $t=2.00 \mathrm{~s}$ to $t=3.00 \mathrm{~s} ;$ (b) the instantaneous velocity at $t=2.00 \mathrm{~s} ;(\mathrm{c})$ the instantaneous velocity at $t=3.00 \mathrm{~s} ;(\mathrm{d})$ the instantaneous velocity at $t=2.50 \mathrm{~s} ;$ and $(\mathrm{e})$ the instantaneous velocity when the particle is midway between its positions at $t=2.00 \mathrm{~s}$ and $t=3.00 \mathrm{~s}$ (f) Graph $x$ vs. $t$ and indicate your answers graphically.

Key Concepts

Derivative in Calculus

The derivative is a fundamental concept in calculus that represents the rate of change of one quantity with respect to another. In the context of motion, it is used to determine instantaneous quantities like velocity (the first derivative of position) and acceleration (the second derivative of position).

Graphical Analysis in Kinematics

Graphical analysis involves plotting the position of an object as a function of time to visually examine its motion. Key aspects include identifying the slope of the position-time curve, which corresponds to instantaneous velocity, and using the graphical representation to aid in understanding and interpreting the behavior of the moving particle.

Instantaneous Velocity

Instantaneous velocity is defined as the rate at which an object's position is changing at a specific moment in time. It is obtained by taking the derivative of the position function with respect to time, reflecting the precise speed and direction of the particle at that instant.

Average Velocity

Average velocity is the overall change in position divided by the total elapsed time over a given interval. It provides a measure of how fast and in what direction an object traveled over that period, regardless of variations in its instantaneous motion.

Position Function in Kinematics

A position function describes how the location of a particle changes with time. It forms the basis of kinematics, allowing for the determination of other motion quantities such as velocity and acceleration by using calculus.

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