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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
Question 9
A particle travels along a curve from O to E in the xy-plane, as shown in the diagram. The position vector of the particle is r, its velocity is v, and its accelera... show full transcript
Worked Solution & Example Answer: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
Step 1
r ⋅ v ≤ 0 and a ≥ 0
Answer
This indicates that the position vector and the velocity vector of the particle are either perpendicular or the particle is standing still (at maximum distance when slowing down). However, the condition a ≥ 0 does not properly represent the slowing down of the particle.
Step 2
r ⋅ v ≤ 0 and a ⋅ v ≥ 0
Answer
This condition suggests that either the particle is not moving away from the origin or is moving towards it, and the acceleration points in the same direction as the velocity vector. As the particle is always slowing down, this indicates the motion is inconsistent with the described behavior.
Step 3
Step 4
r ≥ 0 and a ⋅ v ≤ 0
Answer
This condition is consistent with a particle that is slowing down. The positive position vector indicates that it is away from the origin, while the negative dot product of acceleration and velocity indicates that the acceleration acts opposite to the velocity, leading to a decrease in speed over time.