Instantaneous velocity Consider the position function...

Instantaneous velocity Consider the position function $\left.s(t)=-4.9 t^{2}+30 t+20 \text { (Exercise } 10\right) .$ Complete the following table with the appropriate average velocities. Then make a conjecture about the value of the instantaneous velocity at $t=2.$ $$\begin{array}{|l|l|l|l|l|l|} \hline \begin{array}{l} \text { Time } \\ \text { interval } \end{array} & {[2,3]} & {[2,2.5]} & {[2,2.1]} & {[2,2.01]} & {[2,2.001]} \\ \hline \begin{array}{l} \text { Average } \\ \text { velocity } \end{array} & & & & \\ \hline \end{array}$$

Instantaneous velocity Consider the position function $\left.s(t)=-4.9 t^{2}+30 t+20 \text { (Exercise } 10\right) .$ Complete the following table with the appropriate average velocities. Then make a conjecture about the value of the instantaneous velocity at $t=2.$
$$\begin{array}{|l|l|l|l|l|l|}
\hline \begin{array}{l}
\text { Time } \\
\text { interval }
\end{array} & {[2,3]} & {[2,2.5]} & {[2,2.1]} & {[2,2.01]} & {[2,2.001]} \\
\hline \begin{array}{l}
\text { Average } \\
\text { velocity }
\end{array} & & & & \\
\hline
\end{array}$$

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Key Concepts

Position Function

A position function describes the location of an object as a function of time. It is a key concept in kinematics that gives the object's spatial coordinates at any specific moment. The behavior of the position function can reveal important information about the movement of the object, such as whether it is accelerating or decelerating.

Average Velocity

Average velocity is defined as the change in position divided by the change in time over a specific time interval. This concept provides a measure of the overall speed and direction of an object over that interval, helping to quantify its movement between two points in time.

Instantaneous Velocity

Instantaneous velocity represents the velocity of an object at a precise moment in time. It is determined by considering the limit of the average velocity as the time interval approaches zero, effectively capturing the object's speed and direction at a single point rather than over an interval.

Derivative

The derivative is a fundamental concept in calculus that provides a way to compute the instantaneous rate of change of a function. In the context of motion, the derivative of the position function with respect to time gives the instantaneous velocity. This limit-based approach underpins the transition from average to instantaneous quantities.

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