An autonomous delivery robot begins at the origin of an -coordinate system

PHYSICS · Multi-part · CBSE · Physics · Motion in a Plane · Motion in a Plane

Problem

An autonomous delivery robot begins at the origin of an

xy

-coordinate system. Its programmed route consists of three straight-line segments executed in the following order: • Segment 1: travels

120\,\text{m}

at an angle

\theta

north of east, where

0^{\circ}\le \theta\le 90^{\circ}

. • Segment 2: travels

50\,\text{m}

due west. • Segment 3: travels

30\,\text{m}

due south. (a) Derive an algebraic expression for the robot’s final

x

-coordinate in terms of

\theta

. Clearly indicate the sign of each term. (b) Derive a corresponding expression for the robot’s final

y

-coordinate in terms of

\theta

. (c) Determine the value of

\theta

(if any) for which the robot’s final position lies directly north of its starting point (that is, for which the final

x

-coordinate equals zero). Show all steps leading to your answer. (d) For the value of

\theta

found in part (c), calculate the magnitude of the robot’s displacement from the origin.