# Homework Help: Kinetic Energy Interpreted as Line Integral?

1. Oct 14, 2009

### merryjman

1. The problem statement, all variables and given/known data

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 = t3/2, then its kinetic energy is proportional to:
(a) t2
(b) t3/2
(c) t
(d) t1/2
(e) t0 (i.e. kinetic energy is constant)

2. Relevant equations

velocity is time derivative of position; therefore v $$\propto$$ t1/2

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

3. 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?

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Oct 14, 2009

### 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?

3. Oct 15, 2009

### 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