A delivery robot moves along a straight corridor according to the velocity function…

MATHEMATICS · Multi-part · AP · AP Precalculus · Functions with Parameters, Vectors, and Matrices · Functions with Parameters, Vectors, and Matrices

Problem

A delivery robot moves along a straight corridor according to the velocity function v(t)={⟨0.8,0⟩,0≤t<3,⟨0,0⟩,3≤t<4,⟨−0.6,0⟩,4≤t≤9,\mathbf v(t)= \begin{cases} \langle 0.8,0\rangle, & 0\le t<3,\\[4pt] \langle 0,0\rangle, & 3\le t<4,\\[4pt] \langle -0.6,0\rangle, & 4\le t\le 9, \end{cases} where tt is measured in seconds and velocity components are in meters per second. The robot has mass 1212 kg. The vector ⟨1,0⟩\langle 1,0\rangle points toward the loading dock. Answer the following questions concerning the interval 0≤t≤90\le t\le 9. (a) Calculate the robot’s net displacement vector and the total distance it travels. (b) Determine the robot’s average velocity vector and its average speed. (c) The robot is programmed to return to its starting point exactly at t=9t=9. Verify whether this requirement is satisfied, and justify your conclusion numerically. (d) Compute the robot’s average kinetic energy (in joules) on 0≤t≤90\le t\le 9. Recall that the kinetic energy at time tt is 12m∥v(t)∥2\tfrac12 m\lVert\mathbf v(t)\rVert^{2}. Show all steps leading to your result.

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