Uniform circular motion (kinematics) Transitional Reasoning Download PDF

Questions

1. 1.

   A robotics researcher evaluates the navigation module of an autonomous rover undergoing uniform circular motion at constant speed $v$ on a circular track of radius $R$. The dead-reckoning module records the rover's straight-line displacement and average velocity between two telemetry checkpoints separated by a central angle of $90^\circ$. What is the ratio of the magnitude of the rover's average velocity, $|\vec{v}_{\text{avg}}|$, to its constant speed, $v$, over this interval?

   • A) $\frac{2\sqrt{2}}{\pi}$
   • B) $\frac{\sqrt{2}}{\pi}$
   • C) $\frac{4}{\pi}$
   • D) $\frac{\sqrt{2}}{2}$
2. 2.

   A flight computer monitors an autonomous drone performing uniform circular motion with constant speed $v$ along a horizontal circular track of radius $R$. The system records the drone's motion as it travels between two waypoints that subtend a central angle of $60^\circ$ with respect to the center of the path. What is the ratio of the magnitude of the drone's average acceleration over this interval, $|\vec{a}_{\text{avg}}|$, to the magnitude of its instantaneous centripetal acceleration, $a_c$?

   • A) $\frac{3}{\pi}$
   • B) $\frac{2}{\pi}$
   • C) $\frac{3\sqrt{3}}{\pi}$
   • D) 1
3. 3.

   An engineer calibrates an automated optical inspection sensor that travels in uniform circular motion along a circular guide rail of radius $r$ with constant speed $v$ and period $T$. To inspect a larger component, the guide rail radius is increased to $2r$ while adjusting the motor so that the magnitude of the sensor's centripetal acceleration remains unchanged. What are the sensor's new period $T'$ and new speed $v'$ in terms of $T$ and $v$?

   • A) $T' = \sqrt{2}T$ and $v' = \sqrt{2}v$
   • B) $T' = 2T$ and $v' = \sqrt{2}v$
   • C) $T' = \sqrt{2}T$ and $v' = 2v$
   • D) $T' = 2T$ and $v' = 2v$