A particle moves in a straight line with the given velocity...

A particle moves in a straight line with the given velocity (in meters per second). Find the displacement and distance traveled over the time interval, and draw a motion diagram like Figure 3 (with distance and time labels).
$$
v(t)=36-24 t+3 t^{2}, \quad[0,10]
$$

Key Concepts

Motion Diagram

A motion diagram is a graphical representation of an object's positions at various times along its path. It aids in visualizing the movement, showing changes in position, and marking key times and distances, which help in understanding the overall motion described by the velocity function.

Distance Traveled

Distance traveled is the total length of the path taken by the object, regardless of direction. To compute it, one must integrate the absolute value of the velocity function over the given time interval, ensuring that all segments where the object moves in opposite directions contribute positively to the total.

Turning Points

Turning points occur when the velocity function equals zero, indicating a change in the direction of motion. Identifying these points is essential for accurately computing the distance traveled because the sign of the velocity determines whether integration should be performed on the original function or its absolute value in different intervals.

Displacement

Displacement is the overall change in position from the starting point to the ending point of an object's journey. It is computed by integrating the velocity function over the time interval in question and takes into account the direction of motion.

Velocity Function

A velocity function describes the rate of change of position with respect to time in kinematics. It can be used to determine how fast and in which direction an object is moving at any given instant.

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