
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
A charged particle with charge  $q$  and mass  $m$  is given an initial kinetic energy  $K_0$  at the middle of a uniformly charged spherical region of total charge  $Q$  and radius  $R$ .  $q$  and  $Q$  have opposite signs. The spherically charged region is not free to move. Throughout this problem consider electrostatic forces only.

[asy]
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps);
size(100);
filldraw(circle((0,0),1),gray(.8));
draw((0,0)--(0.5,sqrt(3)/2),EndArrow);
label(" $R$ ",(0.25,sqrt(3)/4),SE);
[/asy]

(a) Find the value of  $K_0$  such that the particle will just reach the boundary of the spherically charged region.

(b) How much time does it take for the particle to reach the boundary of the region if it starts with the kinetic energy  $K_0$  found in part (a)?

Present the answer in LaTex format: \boxed{Your answer}
user