Kinetic Energy Interpreted as Line Integral?

[merryjman]

Homework Statement

From the 1984 Ap Physics C Mechanics Exam: If a particle moves in such a way that its position is described as a function of time by x = t^3/2, then its kinetic energy is proportional to:
(a) t²
(b) t^3/2
(c) t
(d) t^1/2
(e) t⁰ (i.e. kinetic energy is constant)

Homework Equations

velocity is time derivative of position; therefore v \propto t^1/2

Kinetic Energy is proportional to v²; therefore KE \propto t

The Attempt at a Solution

My question deals with not how to obtain the answer, but an interesting question one of my students asked me. He is fresh out of a summer college course in multivariable calculus, and loves to think of everything in terms of line, path and surface integrals now :) He asked me a question I couldn't answer, and I'll try to reproduce it here. He said that, by thinking about this question in terms of line integrals, the implication here is that the kinetic energy is equal/proportional to the arclength of the position curve.

First of all, is this true?

Second, if it is true, does this fact have any physical significance?

Thanks in advance

 

[kuruman]

I am not sure what you mean by the "arclength of the position curve". Is the "position curve" the path that the particle follows? This is a one-dimensional problem, so everything is along a straight line and, in this case, x (arclength?) is not proportional to kinetic energy because they have a different time dependence. What line integral is your student thinking of?

 

[ehild]

The change of KE can be interpreted as a line integral of force, according to the work-energy theorem: The change of the kinetic energy of a point mass while it moves from point A to point B is equal to the work of the resultant force acting on it.

ehild