One-shot Monostable 555 with long trigger

C1 is the capacitor that is used along with R1 to set the pulse length using the formula you gave. So you can substitute R1 for R, and C1 for C in the formula.

The CONTROL lead is used to adjust the interior comparator levels, in this case it is not used. The capacitor C2 just provides some noise immunity to prevent false triggering. It is typically 10 nF to 100 nF.

The output will be equal to V1 when triggered, and ground otherwise.

Instead of using a separate V2 voltage, you can just tie R2 to V1. The TRIGGER voltage just needs to be above V1/3 when not active, but there is no reason it can't be equal to V1. A good value of R2 is 10K.

You should also put a 100 nF capacitor between the Vcc pin and ground.

Here is a simplified view of interior circuit of the 555:

Note the three 5K resistors on the left that create a voltage divider; that's where the name 555 comes from. The resistors set up a voltage of 2/3 V on the - input to the upper comparator C\$_{A}\$, and 1/3 V on the + input of the lower comparator C\$_{B}\$.

When the TRIGGER falls below 1/3 V, the lower comparator C\$_{B}\$ outputs a high and sets the flipflop, and the OUTPUT goes high. The external capacitor C1 also starts to charge. When the external RC network made up of R1 and C1 reaches 2/3 V, the upper comparator C\$_{A}\$ goes high, and resets the flip-flop, and the OUTPUT goes back to 0.

Potential problem: Looking at the interior circuit of the 555, if the TRIGGER input is held low for longer than pulse length, it will keep the lower comparator C\$_{B}\$ high and the OUTPUT will remain high.

You can get around this problem using a differentiating input:

It generates a short negative going pulse regardless of how long you hold the switch down.

Connect the switch to the reset input of the 555, and connect the trigger through a small RC delay:

^{simulate this circuit – Schematic created using CircuitLab}

The 555 is held in reset as long as the switch isn't pushed. When the switch is held in, the 555 goes through its normal timing cycle once, but it aborts if the switch is released early.