91Ó°ÊÓ

At the heart of understanding motion in physics is the position function, which describes how an object's position changes over time. In the case of our bird flying due east, the position function is given by
\( x(t) = 28.0\text{ m} + (12.4\text{ m/s})t - (0.0450\text{ m/s}^3)t^3 \).
This equation encapsulates the entire motion of the bird with respect to time, accounting for an initial position, constant velocity, and acceleration effects due to, for example, wind resistance or changes in the bird's speed.

The first term, 28.0 m, represents the initial distance of the bird from the building; it's the starting point. The linear term with coefficient 12.4 m/s represents the influence of a constant speed on the position over time. Lastly, the cubic term with coefficient -0.0450 m/s³ illustrates a more complex change in movement, possibly an acceleration component that impacts the bird's speed. Understanding this function is crucial for predicting the bird's location at any given time.

Derivative calculus is a fundamental concept in mathematics that provides us with the tools to describe change. For instance, if we want to know how quickly the bird's position is changing, we refer to its velocity, which is the first derivative of the position function with respect to time.

The process of finding this derivative involves using rules of calculus like the power rule, which states that the derivative of \( t^n \) is \( n \times t^{(n-1)} \). In our example, applying the power rule to each term in the position function \( x(t) \) yields the velocity function:
\( v(t) = \frac{dx}{dt} = 12.4\text{ m/s} - 3 \times 0.0450\text{ m/s}^3 \times t^2 \),
which simplifies to \( v(t) = 12.4\text{ m/s} - 0.135\text{ m/s}^3 \times t^2 \).

By taking the derivative, we've transitioned from describing the bird's location to describing how its speed changes over time, providing insight into its dynamic behaviour.

The velocity function is a real-time snapshot of the bird's speed for any given moment. It expresses the rate at which the bird's position is changing. After finding the derivative of the position function, we obtained the velocity function:\( v(t) = 12.4\text{ m/s} - 0.135\text{ m/s}^3 \times t^2 \).

To calculate the instantaneous velocity at a specific time, we substitute the desired time into our velocity function. For time \( t=8.00\text{ s} \), we calculate:
\( v(8.00\text{ s}) = 12.4\text{ m/s} - 0.135\text{ m/s}^3 \times (8.00\text{ s})^2 \), which simplifies to \( v(8.00\text{ s}) = 12.4\text{ m/s} - 0.135\text{ m/s}^3 \times 64 = 4.24\text{ m/s} \).

This value is the bird's instantaneous velocity at the eighth second of its flight. It is essential for understanding movements at an exact moment and differs from average velocity, which would consider the total displacement and time elapsed. Real-world applications of this could be determining the exact moment to photograph the bird in flight or predicting its position for collision avoidance systems.