If this is truly a project for industry, and not a student exercise, then there's lots more things to consider than just the order of startup.
Typically in a conveyor system, the LAST conveyor is started FIRST (otherwise, you'll be feeding product onto a non-moving conveyor, causing jams, etc).
You usually take the Aux contact from the motor starter as an input to the PLC, to confirm that the motor is running. This signal is used to both drive the startup timer, AND to shut down IMMEDIATEDLY any upstream conveyors.
You will often have two types of stop methods in your system: a "controlled shutdown" AND an "immedieate stop". The immediate stop would be for E-stops and motor faults. The controlled shutdown is where you have your cascaded timers (shutting down the FIRST conveyor FIRST, after a delay to give it enough time to get all the product onto the second conveyor, THEN starting the second conveyor shutdown timer (which again allows that last box to move onto the third conveyor, etc.)
ChuckM said:
IMHO, I'd say make your e-stops hardwired, not logical in the PLC.
While I agree with you that the e-stops should be hard wired, it doesn't hurt to ALSO let the PLC know that the E-Stop has been pressed, and for the PLC to shut the system down in parallel. Otherwise, the PLC will be trying to drive the output, it won't see the Aux contact, and assume that the motor FAILED. Or, worse, the PLC will not stop driving the outputs, and so ALL conveyors will be restarted as soon as the E-stop RESET button is pressed. This is probably undesireable.