91Ó°ÊÓ

The concept of the limit of a function is essential in calculus and analyses how a function behaves as its input approaches a certain value. In the context of motion, it is used to describe how the value of a function changes as time goes on and is particularly useful when dealing with quantities that change instantaneously, such as velocity.

For our particle moving along the x-axis, the position is given by the function x(t) = 3t^2. To determine the instantaneous velocity, we examine what happens to the change in position, \(\Delta x\), as the change in time, \(\Delta t\), gets arbitrarily small. This is where the limit comes into play:

\[\lim_{\Delta t \to 0} \frac{\Delta x}{\Delta t}\]
This expression signifies that as \(\Delta t\) approaches zero, we look at the corresponding values of \(\Delta x / \Delta t\) to find out the particle's velocity at a specific instant in time, which is the derivative of the position function with respect to time. In summation, limits allow us to precisely determine instantaneous rates of change when we cannot compute them directly due to the infinitely small time intervals involved.

Understanding the position-time relationship is crucial when studying the motion of an object. The position function, which relates position and time, tells us where an object will be at any particular moment. This relationship is fundamental in physics as it helps in predicting the future positions of moving objects.

For example, in our exercise, the position function is defined as x(t) = 3t^2. This quadratic relationship indicates that the particle's position is proportional to the square of the time. As time increases, the position changes at an increasing rate, which is characteristic of accelerated motion. When we know the time, we can easily compute the particle's position by substituting the value of time into the position function.

A careful analysis of the position-time graph, which would be a parabola opening upwards in this case, also gives insights into the object's speed and direction of motion. Thus, the position-time relationship is an indispensable tool for analyzing motion and understanding how different factors like velocity and acceleration are interconnected.

Velocity, a vector quantity, represents the rate of change of position. It has both a magnitude (speed) and a direction. To understand motion, it is important to distinguish between average and instantaneous velocities.

Average velocity is calculated over a finite interval of time and is defined as the total displacement divided by the total time taken. Mathematically, for a time interval from \(t_1\) to \(t_2\), the average velocity is:

\[v_{\text{avg}} = \frac{x(t_2) - x(t_1)}{t_2 - t_1}\]
However, instantaneous velocity refers to the velocity at a specific point in time. Instead of looking at time intervals, we think of taking a snapshot of the object's velocity at a precise moment. It is the limit of the average velocity as the time interval becomes infinitesimally small, reducing to the first derivative of the position with respect to time:

\[v(t) = \lim_{\Delta t \to 0} \frac{\Delta x}{\Delta t}\]
In our exercise, by finding the limit of \(\Delta x/\Delta t\) as \(\Delta t\) approaches zero, we determine the instantaneous velocity at \(t = 3.00 s\), which reflects the true velocity of the particle at that exact moment. This concept provides a more precise and relevant description of an object's motion than the average velocity.