Which of the following assignments of value to the variable $p$ makes the family of functions $f_t:[0,1]\rightarrow\mathbb R^2$ sending:

$$s\mapsto((1-t)\cos(2\pi s),(1-t)+(1-tp)\sin(2\pi s))$$

...where $0\le t\le 1$ a homotopy of a loop to a point at the origin?