On the menu bar try clicking on Evaluation, Quit Kernel, Local, Quit
and then try evaluating your two problems again.
In[1]:= Solve[{
c^2 (F1 - m*g*a1)/(g^3*v1*m) == v1*a1 + v2*a2 + v3*a3,
c^2 (F2 - m*g*a2)/(g^3*v2*m) == v1*a1 + v2*a2 + v3*a3,
c^2 (F3 - m*g*a3)/(g^3*v3*m) == v1*a1 + v2*a2 + v3*a3},
{a1, a2, a3}]
Out[1]= {{
a1 -> -((-c^2 F1 + F2 g^2 v1 v2 - F1 g^2 v2^2 + F3 g^2 v1 v3 - F1 g^2 v3^2)/
(g m (c^2 + g^2 v1^2 + g^2 v2^2 + g^2 v3^2))),
a2 -> -((-c^2 F2 - F2 g^2 v1^2 + F1 g^2 v1 v2 + F3 g^2 v2 v3 - F2 g^2 v3^2)/
(g m (c^2 + g^2 v1^2 + g^2 v2^2 + g^2 v3^2))),
a3 -> -((-c^2 F3 - F3 g^2 v1^2 - F3 g^2 v2^2 + F1 g^2 v1 v3 + F2 g^2 v2 v3)/
(g m (c^2 + g^2 v1^2 + g^2 v2^2 + g^2 v3^2)))}}
In[2]:= Solve[{
fx == m*vx*g^3 (vx*ax + vy*ay + vz*az)/c^2 + g*m*ax,
fy == m*vy*g^3 (vx*ax + vy*ay + vz*az)/c^2 + g*m*ay,
fz == m*vz*g^3 (vx*ax + vy*ay + vz*az)/c^2 + g*m*az},
{ax, ay, az}]
Out[2]= {{
ax -> -((-c^2 fx + fy g^2 vx vy - fx g^2 vy^2 + fz g^2 vx vz - fx g^2 vz^2)/
(g m (c^2 + g^2 vx^2 + g^2 vy^2 + g^2 vz^2))),
ay -> -((-c^2 fy - fy g^2 vx^2 + fx g^2 vx vy + fz g^2 vy vz - fy g^2 vz^2)/
(g m (c^2 + g^2 vx^2 + g^2 vy^2 + g^2 vz^2))),
az -> -((-c^2 fz - fz g^2 vx^2 - fz g^2 vy^2 + fx g^2 vx vz + fy g^2 vy vz)/
(g m (c^2 + g^2 vx^2 + g^2 vy^2 + g^2 vz^2)))}}
It is likely, but impossible to verify now, that you had some value assigned to something which gave no solution to the second problem.
This is a somewhat common problem and often extremely difficult to recognize when it happens unless you know the result is incorrect.
In the future, a powerful method for checking this is to scrape-n-paste your three equations into a new cell and tapping <shift><enter>.
If the result appears to be exactly what your system was then it is unlikely that something had accidentally been assigned to some variable.
But if the result looks somewhat or very different then you can try to track down which variable might have have some unexpected value.
Or perhaps it is a difference in behavior between one version of Mathematica and another.
