91Ó°ÊÓ

The position function is crucial in understanding motion along a path. In kinematics, it defines the location of an object in space at any given time. For the problem presented, the position function is given as:\[ x = 3t - 4t^2 + t^3 \]where \( x \) denotes the position in meters and \( t \) is the time in seconds. This function helps us pinpoint where exactly the object is located along the x-axis at any specified time.To evaluate the object's position at different times, simply substitute the time value \( t \) into the position function and solve for \( x \). This gives you the exact position of the object at that particular moment.

Displacement is a key concept in understanding how far an object has moved over a time period, regardless of its actual path. It is defined as the change in position from the initial point to the final point.In our example, to find the displacement from \( t = 0 \) to \( t = 4 \), calculate the initial position at \( t = 0 \):- Plugging \( t = 0 \) into the position function yields \( x_0 = 0 \)And the final position at \( t = 4 \):- Calculating at \( t = 4 \) gives \( x_4 = 12 \)Thus, the displacement can be found using:\[ \Delta x = x_4 - x_0 = 12 - 0 = 12 \text{ meters} \]The displacement is always a straight-line measure from start to finish and does not concern the path taken.

Average velocity is a measure of how an object's position changes over a specified time interval. It's defined as the total displacement divided by the total time taken.For the interval from \( t = 2 \) to \( t = 4 \):- The positions are \( x_2 = -2 \) meters and \( x_4 = 12 \) meters respectively.- The displacement \( \Delta x \) is \( 12 - (-2) = 14 \) meters- Time interval \( \Delta t \) is \( 4 - 2 = 2 \) secondsHence, the average velocity \( v_{avg} \) can be calculated as:\[ v_{avg} = \frac{\Delta x}{\Delta t} = \frac{14}{2} = 7 \text{ m/s} \]Always remember that average velocity considers only the initial and final positions, not the path or fluctuations between these points.

Graphing motion helps visualize how an object's position changes over time. The position vs. time graph provides a clear picture of the object's journey.To create the graph for our function:- Calculate key positions at various time points: - At \( t = 0 \), \( x = 0 \) - At \( t = 1 \), \( x = 0 \) - At \( t = 2 \), \( x = -2 \) - At \( t = 3 \), \( x = 0 \) - At \( t = 4 \), \( x = 12 \)Plot these points on a graph and connect them with a smooth curve to represent the object's motion.To determine average velocity from \( t = 2 \) to \( t = 4 \) on a graph, the slope of the line connecting these points represents it. The steeper the slope, the higher the average velocity. This graphical representation enhances understanding by providing a visual link to the calculated values.