91Ó°ÊÓ

In kinematics, average velocity is a crucial concept that helps us understand the general behavior of an object over a time interval. It is defined as the total displacement divided by the total time taken. For our problem, the formula for average velocity is given by:\[ v_{\text{avg}} = \frac{x(t_2) - x(t_1)}{t_2 - t_1} \]Here, \( x(t) \) is the position function, which indicates the location of the car at any time \( t \). For the interval from \( t = 0 \) to \( t = 10 \) seconds, we calculated the car's initial and final positions as 0 m and 22800 m, respectively.
- The displacement for this interval is \( 22800 \) m.- The time interval is \( 10 \) seconds.Substituting these values, we find the average velocity \( v_{\text{avg}} = 2280 \) m/s. This means that, on average, the car moved at 2280 meters per second over the 10-second period.

Instantaneous velocity refers to the velocity of an object at a specific instant. It is given by the derivative of the position function, capturing the rate of change of position at any point in time. In this problem, the position function \( x(t) = bt^2 - ct^3 \) is differentiated to obtain the velocity function:\[ v(t) = \frac{d}{dt}[bt^2 - ct^3] = 2bt - 3ct^2 \]Simplifying this with the given constants \( b = 240 \, \text{m/s}^2 \) and \( c = 0.120 \, \text{m/s}^3 \), we have:\[ v(t) = 480t - 0.360t^2 \]This expression allows us to calculate the car's velocity at any given time.

• At \( t = 0 \), the velocity is \( 0 \) m/s, indicating that the car starts from rest.
• At \( t = 5 \) seconds, the velocity is \( 2391 \) m/s.
• Finally, at \( t = 10 \) seconds, the velocity is \( 4764 \) m/s.

These values show how the velocity changes over time, confirming the car's acceleration and deceleration behavior within the given timeframe.

A position function in kinematics describes how the position of an object changes with respect to time. It is a key tool for understanding motion, as it models the trajectory of the object. In this exercise, the position function is given by:\[ x(t) = b t^2 - c t^3 \]where \( b = 240 \, \text{m/s}^2 \) and \( c = 0.120 \, \text{m/s}^3 \). This quadratic-cubic polynomial indicates the car's displacement from the starting point. At different time points:

• When \( t = 0 \), the position \( x(0) = 0 \) m, meaning the car is at the traffic light.
• After 10 seconds, \( x(10) = 22800 \) m, representing the position of the car far from the start.

The position function provides valuable information about the path traveled and helps in predicting future positions with given parameters.

The velocity function is derived by differentiating the position function with respect to time. It provides detailed insights into how the speed of an object changes over time. From our problem, the velocity function was identified as:\[ v(t) = 480t - 0.360t^2 \]This expression succinctly captures the car’s velocity dynamics, indicating how quickly the car changes speed.

• The coefficient \( 480 \) reflects the car's initial acceleration rate.
• The term \( -0.360t^2 \) shows the effect of deceleration due to cubic speed increase in distance.

By solving \( v(t) = 0 \), we determined the times when the car was at rest, finding these moments at \( t = 0 \) and \( t = 1333.33 \) seconds. This highlights critical points in time where the velocity, and thus motion, ceases, which is crucial for understanding the complete motion behavior of the car. Understanding the velocity function helps in predicting motion events and planning accordingly.