A particle moves along a straight line with a differentiable velocity function…

MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration

Problem

A particle moves along a straight line with a differentiable velocity function v(t)v(t) measured in meters per second for 0≤t≤80\le t\le 8. Two students analyze the motion over this time interval. • Student A evaluates ∫08v(t) dt\displaystyle \int_{0}^{8} v(t)\,dt and obtains \!−2-2 meters. • Student B evaluates ∫08∣v(t)∣ dt\displaystyle \int_{0}^{8} |v(t)|\,dt and obtains \!1010 meters. (a) Interpret Student A’s result in the context of the particle’s motion. Be sure to include units. (b) Interpret Student B’s result in the context of the particle’s motion. Explain why Student B’s integral cannot be negative. (c) Given that the displacement is −2-2 m and the distance traveled is 1010 m, describe qualitatively how the particle’s position changes on 0≤t≤80\le t\le 8.

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