A particle moves in the plane with continuously differentiable position vector ,
PHYSICS · Multi-part · A Level · Physics · Mechanics and materials · Mechanics and materials
Problem
A particle moves in the plane with continuously differentiable position vector r(t), 0≤t≤6. During the motion the particle never reverses direction along its path.
Answer the following questions about average quantities associated with the motion. Express any vector quantities using angle–bracket notation ⟨⋅,⋅⟩ and include appropriate units.
(a) Explain the difference between the particle’s average velocity on the time interval 0≤t≤6 and its average speed on the same interval. In your explanation, identify how each quantity is computed and describe what physical information each conveys.
(b) Suppose r(0)=⟨2,1⟩\,m and r(6)=⟨−4,10⟩\,m. The total distance traveled by the particle on 0≤t≤6 is 15 m. Determine the magnitude of the particle’s average velocity vector and its average speed. Compare the two numerical values and clearly interpret the comparison in the context of the motion.
(c) Describe a plausible qualitative shape of the trajectory that would make the particle’s average speed strictly greater than twice the magnitude of its average velocity on 0≤t≤6. Justify how your description produces the required inequality.