(II) The position of a ball rolling in a straight line is...

(II) The position of a ball rolling in a straight line is given by $x=2.0-3.6 t+1.1 t^{2},$ where $x$ is in meters and $t$ in seconds. $(a)$ Determine the position of the ball at $t=1.0 \mathrm{s}$ $2.0 \mathrm{s},$ and 3.0 $\mathrm{s}$ (b) What is the average velocity over the $2.0 \mathrm{s},$ and 3.0 $\mathrm{s}$ (b) What is the average velocity over the interval $t=1.0 \mathrm{s}$ to $t=3.0 \mathrm{s} ?$ (c) What is its instanta- neous velocity at $t=2.0 \mathrm{s}$ and at $t=3.0 \mathrm{s} ?$

(II) The position of a ball rolling in a straight line is given by
$x=2.0-3.6 t+1.1 t^{2},$ where $x$ is in meters and $t$ in
seconds. $(a)$ Determine the position of the ball at $t=1.0 \mathrm{s}$
$2.0 \mathrm{s},$ and 3.0 $\mathrm{s}$ (b) What is the average velocity over the
$2.0 \mathrm{s},$ and 3.0 $\mathrm{s}$ (b) What is the average velocity over the
interval $t=1.0 \mathrm{s}$ to $t=3.0 \mathrm{s} ?$ (c) What is its instanta-
neous velocity at $t=2.0 \mathrm{s}$ and at $t=3.0 \mathrm{s} ?$

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Key Concepts

Derivative

A core concept in calculus, the derivative measures how a function changes as its input changes. In mechanics, taking the derivative of a position function with respect to time yields the instantaneous velocity, linking mathematical analysis to the study of motion.

Instantaneous Velocity

This is the velocity of an object at a specific moment in time. It is defined as the derivative of the position function with respect to time, providing a precise rate of change of position at that exact instant.

Position Function

This is the mathematical expression that describes the object's location with respect to time. It encapsulates the kinematics of the moving object, allowing one to compute its position at any given instant by substituting the time value into the function.

Average Velocity

This concept represents the overall change in position divided by the total time elapsed over a specific interval. It provides a measure of the object's general speed and direction of motion between two time points, regardless of variations in speed within that interval.

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