Final answer:
The question requires finding the instantaneous velocity of an object moving in a straight line, which is calculated by taking the derivative of the position function, and the average velocity over a certain time period, calculated by the change in position over time.
Step-by-step explanation:
The subject of this question is Physics, and it is a typical problem found at the High School level, particularly in a physics class that covers kinematics. The task is to calculate the instantaneous velocity at a given time using a position-time equation and also to find the average velocity over a specific interval.
To find the instantaneous velocity at a time t, we take the derivative of the position function, x(t), with respect to time. The equation given for instantaneous velocity is basically the derivative of the position function. Using the power rule (Equation 3.7), we differentiate each term of the position equation. In example 3.4 where x(t) = 3.0t - 3t², the instantaneous velocity v(t) at time t is v(t) = dx/dt = 3.0 - 6t.
For average velocity, we calculate this by taking the displacement (change in position) divided by the time interval. Using the position function x(t), we find the positions at the beginning and end of the given time interval and then divide this by the time passed to get the average velocity.
To find the velocity vector as in example 4.3, we take the derivative of each component of the position vector with respect to time. The magnitude of the velocity vector at a specific time gives the speed at that time.
To calculate displacement and acceleration, we use the area under the velocity-time graph and the slope of the velocity-time graph, respectively. Understanding these concepts and calculations are fundamental in the study of physics, particularly when analyzing motion.