Provide an appropriate response.
-The following equations each describe the motion of a particle. For which path is the particle's speed constant?
(1) $r ( t ) = t ^ { 3 } i + t ^ { 2 } j$
(2) $r ( t ) = \cos ( 8 t ) i + \sin ( 6 t ) j$
(3) $r ( t ) = t i + t j$
(4) $r ( t ) = \cos \left( - 5 t ^ { 2 } \right) \mathbf { i } + \sin \left( - 5 t ^ { 2 } \right) \mathbf { j }$
A) Path (4) and Path (2)
B) Path (1)
C) Path (2) and Path (3)
D) Path (3)

Question

Quizplus · Accepted Answer

The speed of a particle is given by the magnitude of its velocity vector. Thus, for the particle to have a constant speed, its velocity vector must have a constant magnitude. This occurs when the particle moves at a constant rate along a straight line. Path (3) is a straight line, as it is a linear combination of two constant vectors (i and j), so the particle moves at a constant rate (constant speed) along this path. The other paths involve circular or nonlinear motion, which means that the particle's velocity vector changes direction and therefore has varying magnitude, so the paths do not have constant speed.

Provide an Appropriate Response r(t)=t3i+t2jr ( t ) = t ^ { 3 } i + t ^ { 2 } jr(t)=t3i+t2j

Provide an Appropriate Response $r ( t ) = t ^ { 3 } i + t ^ { 2 } j$