A particle travels along a curve from O to E in the xy-plane, as shown in the diagram - HSC - SSCE Mathematics Extension 2 - Question 9 - 2023 - Paper 1

To determine which option is consistent with the motion of the particle, we examine the implications of the velocity and acceleration given that the particle is slowing down.

1. Understanding Velocity and Acceleration: Since the particle is moving along the curve from O to E and is always slowing down, the acceleration must be opposite to the direction of the velocity vector. This indicates that the dot product of the velocity vector and the acceleration vector should be less than or equal to zero, i.e., a ullet v \\leq 0 This aligns with option D.

2. Dot Product Analysis: Further, for the particle to be moving in the direction indicated, the velocity vector and the position vector must exhibit the following:

   • If r ullet v \\geq 0 This implies that the direction of the particle’s motion is consistent, meaning the speed is maintained as it travels along the increasing position vector from point O to E.

Thus, considering the criteria for slowing down while moving forward along the trajectory, option D aligns with the physical conditions described.