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Q9 (CDS-II/2018) Science & Technology › Basic Science (Physics, Chemistry, Biology) › Basic mechanics concepts Answer Verified

The position vector of a particle is given by r = 2t²î + 3tĵ + 4k̂. Then the instantaneous velocity v and acceleration a respectively lie:

  • (A) on the xy-plane and along the z-direction
  • (B) on the xz-plane and along the x-direction
  • (C) on the yz-plane and along the x-direction
  • (D) on the xy-plane and along the x-direction

Result
Your answer: —  ·  Correct: D

Explanation

The position vector is given as r = 2t²i + 3tj + 4k. To find the instantaneous velocity (v), we take the first derivative of r with respect to time: v = dr/dt = 4ti + 3j + 0k. Since the z-component is zero, the velocity vector lies entirely in the xy-plane. To find the acceleration (a), we take the derivative of the velocity: a = dv/dt = 4i + 0j + 0k [1]. The acceleration vector has only an x-component, meaning it is directed along the x-axis. Therefore, the velocity lies on the xy-plane and the acceleration is along the x-direction. This follows the standard kinematic procedure where velocity and acceleration are successive derivatives of the position vector.

Sources

  1. [1] https://www.maths.dur.ac.uk/users/W.J.Zakrzewski/dynamics/problems_solns.pdf
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