Answer
The fraction of the initial kinetic energy that is stored in the spring is $~~0.8$
Work Step by Step
We can find the total kinetic energy before the collision:
$K_i = \frac{1}{2}(0.060~kg)(22~m/s)^2$
$K_i = 14.5~J$
In part (a), we found that the speed of the spring gun (and ball) after the collision was $4.4~m/s$
We can find the total kinetic energy of the system after the collision:
$K_f = \frac{1}{2}(0.300~kg)(4.4~m/s)^2$
$K_f = 2.9~J$
We can find the change in the kinetic energy of the system:
$\Delta K = K_f - K_i = 2.9~J - 14.5~J = -11.6~J$
The total kinetic energy of the system decreases by $~~11.6~J$
This "missing" energy is stored in the spring.
We can find the fraction of the initial kinetic energy that is stored in the spring:
$\frac{11.6~J}{14.5~J} = 0.8$
The fraction of the initial kinetic energy that is stored in the spring is $~~0.8$
