Final answer:
To estimate the instantaneous velocity at t = 3 seconds, differentiate the position function and evaluate at t = 3. If the position function is r(t) = 3.0t³î, then the instantaneous velocity is v(3 s) = 81î m/s. For average velocity between 1.0 s and 3.0 s, calculate the difference in position and divide by the time interval.
Step-by-step explanation:
To estimate the instantaneous velocity when t = 3 seconds, you would typically take the derivative of the position function and then substitute the given time into the derived velocity function. However, the information given appears to refer to different position functions for different examples, which makes the context unclear. Based on the snippet from the CHECK YOUR UNDERSTANDING section, if we consider the position function r(t) = 3.0t³î, then using calculus to differentiate it with respect to t gives us the velocity function. Taking the derivative of r(t) yields v(t) = 9.0t²î, and substituting t = 3 s into the velocity function gives us v(3 s) = 9.0(3)²î = 81î m/s as the instantaneous velocity.
To calculate the average velocity between 1.0 s and 3.0 s, we would need the position at both times and then divide the difference in position by the time interval.