Final answer:
The position of a particle as a function of time describes its location at any given time, based on the position function r(t). Instantaneous velocity is found by taking the derivative of r(t), while average velocity is determined by the change in position over a time interval.
Step-by-step explanation:
The position of a particle as a function of time is a fundamental concept in physics which describes where a particle is located at any given moment. This is expressed mathematically by a position function r(t). To find instantaneous velocity and acceleration, one must take the first and second derivatives of the position function, respectively.
To find the instantaneous velocity at t = 3 s for a particle with position function r(t) = 3.0t³î + 4.0ç, we take the derivative of r(t) with respect to time to get v(t) = 9.0t²î, and then evaluate at t = 3 s to get v(3.0s) = 81.0î m/s.
The average velocity between 2 s to 4 s is calculated by finding the difference in position at these times and dividing by the time interval. Whereas the instantaneous velocity at t = 3 s is the derivative of the position function evaluated at that moment, which can be different from the average velocity within the interval.