A particle moves along a line with velocity function...

A particle moves along a line with velocity function $v(t)=t^{2}-t,$ where $v$ is measured in meters per second. Find (a) the displacement and (b) the distance traveled by the particle during the time interval $[0,5] .$

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Key Concepts

Displacement through Definite Integration

Displacement is defined as the net change in position of an object over a specific time interval. It is obtained by integrating the velocity function over that interval. The result of this integration gives a signed value, where the sign indicates the direction of motion. This concept emphasizes that displacement is a vector quantity and can be positive, negative, or zero depending on the motion of the object.

Distance Traveled using Integration of Absolute Value

Distance traveled is the total length of the path taken by an object, regardless of the direction of motion. It is calculated by taking the integral of the absolute value of the velocity function over the time interval of interest. This ensures that all portions of the journey, even those where the object reverses direction, contribute positively to the total distance traveled. When the velocity function changes sign, the integration is typically split so that the absolute value is properly accounted for in each segment.

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