Answer
The pebble flies up to a height of $12.8~meters$ above the starting point.
Work Step by Step
By conservation of energy, the gravitational potential energy at maximum height will be equal to the elastic potential energy stored in the catapult initially:
$mgh = \frac{1}{2}kx^2$
$h = \frac{kx^2}{2mg}$
$h = \frac{(320~N/m)(0.20~m)^2}{(2)(0.051~kg)(9.80~m/s^2)}$
$h = 12.8~m$
The pebble flies up to a height of $12.8~meters$ above the starting point.
