Your \$\mathbf{I_L}\$ is correct, but your phase angle for \$\mathbf{U_L}\$ is wrong. An inductor's voltage leads the current by 90 degrees; you subtracted. The voltages are larger than the input because the circuit is reactive, but total power is conserved and you can check that \$\mathbf{U_q} = \mathbf{U_L} + \mathbf{U_C}\$ by cross-verifying \$\mathbf{U_C}=\mathbf{I_L}(\mathbf{Z_C} \parallel R)\$.

Edit: Here are the values I calculated with Python:

```
from math import pi, e
w = 2 * pi * 636.62
R = 0.11e3
ZL = 1j * w * 8.5e-3
ZC = -1j / w / 10e-6
Z = ZL + (R * ZC) / (R + ZC)
U = 220 * e ** (1j * pi * -50 / 180)
I = U / Z
UL = I * ZL
UC = U - UL
IC = UC / ZC
def polar_str(x):
from math import atan2
angle = atan2(x.imag, x.real) / pi * 180
return "({:7.3f}, {:8.3f})".format(abs(x), angle)
print('\n'.join(v + ' = ' + polar_str(eval(v)) for v in
[' R','ZL', 'ZC', ' Z', ' U', ' I', 'UL', 'UC', 'IC']))
output: (mag, phase)
R = (110.000, 0.000)
ZL = ( 34.000, 90.000)
ZC = ( 25.000, -90.000)
Z = ( 11.567, 62.155)
U = (220.000, -50.000)
I = ( 19.019, -112.155)
UL = (646.657, -22.155)
UC = (463.659, 170.649)
IC = ( 18.546, -99.351)
```

You know that the capacitance of two capacitors C1 and C2 in parallel is C1+C2.

You know that the capacitance of two capacitors C1 and C2 in series is
\$ 1 \over { {1 \over C1} + {1 \over C2} }\$

You should be able to find the capacitance if the 7.22uF capacitor was not there, using the first equation.

You should be able to work out the capacitance in series with the unknown capacitance C using a combination of the two equations.

Then just re-arrange the second equation to find the value of the unknown capacitor.

## Best Answer

The force available to accelerate the projectile decreases over time and distance. At the same time, the deceleration forces are increasing (friction with the rails, air resistance, etc.) You need to find the crossover point where the net force drops to zero, and have the rails end somewhat before that point. That's where you'll have the greatest "muzzle velocity".