Find the instantaneous velocity of the particle described in...

Find the instantaneous velocity of the particle described in Figure $\mathrm{P} 2.3$ at the following times: (a) $t=1.0 \mathrm{~s}$,
(b) $t=8.0 \mathrm{~s},(\mathrm{c}) t=4.5 \mathrm{~s}$, and
(d) $t=7.5 \mathrm{~s}$.

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Key Concepts

Position-Time Graph

A position-time graph is a graphical representation of an object's motion where position is plotted on the vertical axis and time on the horizontal axis. The slope of the curve at any point on this graph provides information about the object's instantaneous velocity. A steeper slope indicates a higher velocity, while a horizontal line indicates zero velocity.

Derivative

The derivative is a fundamental concept in calculus that measures how a function changes as its input changes. When applied to motion, the derivative of the position function with respect to time gives the instantaneous velocity. This process involves taking the limit of the average velocity over smaller and smaller time intervals, resulting in the exact rate of change at a precise moment.

Instantaneous Velocity

Instantaneous velocity is the velocity of an object at a specific point in time. It is defined mathematically as the derivative of the position function with respect to time. In practical terms, if you have a position-time graph, the instantaneous velocity at any given time is represented by the slope of the tangent line to the graph at that point.

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