Provide an appropriate response.
-The following equations each describe the motion of a particle. For which path is the particle's velocity vector alv orthogonal to its acceleration vector?
(1) $r ( t ) = t ^ { 5 } i + t ^ { 3 } j$
(2) $r ( t ) = \cos ( 10 t ) \mathbf { i } + \sin ( 9 t ) \mathbf { j }$
(3) $r ( t ) = t i + t ^ { 3 } j$
(4) $r ( t ) = \cos ( 4 t ) i + \sin ( 4 t ) j$
A) Path (2) and Path (4)
B) Path (4)
C) Path (3)
D) Path (1)

Question

Quizplus · Accepted Answer

For the velocity vector to be orthogonal to the acceleration vector, the dot product of the two vectors must equal zero. In other words, the velocity and acceleration vectors must be perpendicular to each other. Using the formulas for velocity and acceleration, we can find that for path (4), the velocity vector is always tangent to the path and the acceleration vector is always perpendicular to the path. Therefore, the velocity vector and the acceleration vector are orthogonal for path (4) only.

Provide an Appropriate Response r(t)=t5i+t3jr ( t ) = t ^ { 5 } i + t ^ { 3 } jr(t)=t5i+t3j

Provide an Appropriate Response $r ( t ) = t ^ { 5 } i + t ^ { 3 } j$