Early American rockets used an RC circuit to set the time for the rocket to begin re-entry after launch (true story). Assume the capacitor is initially charged, and then discharge begins at launch, t = 0 (this was when the umbilical connector was disconnected). Further assume the resistor in the RC circuit is 1k Ω, and the control circuit begins re-entry when the voltage drops ~63% or to ~37% of full-scale (i.e. 1 time constant), and re-entry is desired to commence at 5 minutes (i.e. 300 seconds after launch). Calculate the capacitor value required.

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cjrodriguez89 cjrodriguez89 · Answer 1 · 2020-02-14T14:11:48+01:00

Answer:

C = 0.3 F

Explanation:

The expression for a voltage in a capacitor with an initial voltage V₀, as a function of time, is given by the following equation:

[tex]V= Vo*e^{-t/RC}[/tex]

When V = =.37*V₀, we have the following expression:

[tex]\frac{V}{Vo} = e^{-t/RC} = 0.37[/tex]

Applying ln to both sides, we have:

[tex]\frac{-t}{RC} = ln 0.37 = -1[/tex]

⇒ t = R*C

if t= 300 s, and R = 10³ Ω, we can solve for C as follows:

[tex]C = \frac{t}{R} = \frac{3e2 s}{1e3 ohms} = 0.3 F[/tex]

So, the required value for C is 0.3 F.