The timer circuit operates by controlling the sign of the output voltage of the OP amp. in the beginning, the circuit is connected to the 9V power supplier and both of capacitor and transistor will act like a open circuit at this time. From Kirchoff's Voltage Law, V+ is 4.5 volts because R2 and R3 acts like voltage divider. Due to the the large internal resistance of OP Amp, the current flow through it will be zero, which means V- equal to 9 volts.So Vout is negative. When a low voltage input pulse from the logic circuit triggers the timer circuit, the transistor will become a closed switch and the capacitor will discharge through the transistor loop. The voltage across the capacitor decreases from 9 volts to 0. As V+ > V- , the Vout becomes positive then capacitor starts to recharge and time starts to countdown. For charging the capacitor, R1 controls the current flows into the loop, which means it controls the time period for the timer circuit. We need from the time period from 5s to 10s therefore R1 should be 10kΩ. If the value of R1 increases, the countdown time period also increases. Because as the resistance of R1 increases the current will decreases then the capacitor needs more time to be recharged. Finally, when the voltage across the capacitor reaches 4.5V , Vout will becomes negative again. At this time, the timer circuit sends a signal to the micro-controller to launch the rocket.