Suppose that a particle moves along a straight line with...

Suppose that a particle moves along a straight line with velocity defined by $v(t)=t^{2}-3 t-18, \quad$ where $0 \leq t \leq 6$ (in meters per second). Find the displacement at time $t$ and the total distance traveled up to $t=6$

Key Concepts

Velocity Function

The velocity function represents the rate of change of position with respect to time. It provides both the magnitude and the direction (sign) of the particle's motion along a straight line. Analyzing the velocity function is essential in determining other motion characteristics such as displacement and total distance traveled.

Displacement

Displacement is the net change in position of the particle over a given time interval. It is computed as the definite integral of the velocity function from the initial time to a specified time. Importantly, displacement takes into account the direction of movement, meaning that movements in opposite directions can cancel each other out.

Total Distance Traveled

Total distance traveled measures the entire path length covered by the particle, regardless of direction. Unlike displacement, it is calculated by integrating the absolute value of the velocity function over the time interval. This process involves identifying intervals where the velocity function is negative and accounting for these by taking their absolute contributions.

Sign Changes in Velocity

Identifying the points in time where the velocity function changes sign is crucial when calculating the total distance traveled. These sign changes indicate moments where the particle reverses its direction. By finding these points, the integration for total distance is partitioned into segments where the velocity maintains a consistent sign, allowing for accurate computation of the absolute area under the velocity-time graph.

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