Can we conserve something?

This problem is solved through conservation of energy and angular momentum. By conservation of angular momentum, $v_i r_i = v_f r_f$ , so $v_f = \frac{r_i}{r_f} v_i = \frac{1}{3} v_i$ .

Since the force comes from an elastic cord, there is no potential energy until it is stretched out to 0.9 m, at which point the cord starts stretching, and the potential energy is $U = \frac{1}{2} k (r - 0.9)^2$ .

By conservation of energy,

$\frac{1}{2}m v_{i}^{2} = \frac{1}{2} m v_{f}^{2} - \frac{1}{2} k (r_{f} - 0.9)^2$

Plugging in values and some algebra gives $v_{f} = \frac{3}{2} \frac{\text{m}}{\text{s}}$ .

To find the radius of curvature $r_{c}$ at that point, pretend that the object is moving in a circle with speed $v_{f}$ and has a constant acceleration inward of $\frac{F}{m} = -\frac{k}{m (r_{f} - 0.9)}$ .

$\frac{v_{f}^{2}}{r_{c}} = \frac{k}{m (r_f - 0.9)}$

$r_c = \frac{v_{f}^{2} m (r_f - 0.9)}{k}$

Plugging in values gives $r_{c} = 3.75 \text{cm}$ .

there is a loop in framing of question @avi solanki please add that the component of velocity along string vanishes when length=1.2