Converting displacement to distance

Oct 2010
21
0
Suppose you have the function for a particle's displacement (which is a vector).

Is it possible to manipulate this function so that the result is an expression of the function's total distance (which is a scalar)?

Consider s(t) = 5t is an expression of the particle's displacement.

Intuitively, the distance travelled will also be 5t, but if there are complicated functions that go up and down,

s(t) = sin(t)
where s(t) is the displacement (NOT THE distance),
then how can the function be manipulated to give the absolute value of the actual distance covered by the particle?

Feel free to use calculus to express the answer and forgive the lack of LaTeX.