Final answer:
To determine the instantaneous velocity and speed of a particle, one must take the derivative of the given position function and the absolute value of that derivative, respectively. The times at which the particle stops can be found by setting the velocity function to zero and solving for time.
Step-by-step explanation:
The question pertains to the instantaneous velocity and speed of a particle as functions of time, which is a concept in physics specifically dealt with in the study of kinematics. To find the instantaneous velocity, one must calculate the derivative of the position function, r(t). Instantaneous velocity is a vector that can be positive or negative, depending on the direction of motion.
On the other hand, instantaneous speed, which is a scalar quantity, is the absolute value of instantaneous velocity and is always positive. If the velocity is zero at any point, the particle momentarily stops, which can be determined by setting the velocity function equal to zero and solving for time.
For a given position function r(t), such as r(t) = 2.0t²î + (2.0 + 3.0t)ç + 5.0t km, you would find the instantaneous velocity by differentiating r(t) with respect to time t. This gives you the velocity function v(t). To find the instantaneous speed, you would then take the absolute value of the velocity vector components. By setting the components of v(t) to zero and solving for t, you can find the times at which the particle stops.