The problem with hobby related solutions is documentation is limited and not spec'd like commercial components or modules.
It is possible that it may work but make a block wiring diagram and consult with the OEM is advised. Mind you I don't know if they have adequate support for your question as these are built in China.
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It appears that your ultrasonic load is TTL so there is no problem switching 5 at the same time with a BEC, but I wonder if you have considered the effects of crosstalk on firing all at the same time. They indicate a 15deg detection angle but this would depend on the reflection angle of objects you wish to detect. There may be phasing issues with reflection cancellations like having 5 tweeters directed in a room. Reading the response of each echo in parallel with a time interval count won't be a simple textbook result with non-smooth objects with 5 senders.
YOu can test your orthogonal array design with any signal pulse generator and look at the signal on a parallel port logic analyzer or scope to ensure what you are design will work.
Power drive is the least of your concerns from these low power devices. Noise avoidance from conducted and radiated sources will be paramount and design of the transponder array must come first. I would spend some time on testing this part 1st to identify all the electrical, physical, acoustic, EMI, thermal, vibration both conducted and radiated sources of interference and how each affects your SONAR expectations with different objects. Will it be microphonic with vibration or loud pulse noises. How well does it reject other ultrasound sources of noise? Will the TTL Echo output change in pulse width with signal strength or just the delay time.
Will you get echos from the wrong sender due to corner refection effects.
Likely the contacts will weld the 1st time you use it.
2nd, the pull-in time of the relays will vary (nearly randomly), and this will degrade the sequencing of your coil gun.
3rd -- the EM pulse from the coil gun might cause the Arduino to malfunction. You'll have to shield it very well from the discharge -- the relays might not allow that.
Best to use semiconductors -- IGBTs would be well suited for this task, and can run 1000's of A
Best Answer
OK, now at least we know what you're talking about.
They are compatible in the way that you can directly connect the Arduino output to the relay module's input. Arduino's logic is 5 V, and the module needs 2.5 to 20 V input to drive the relay. So that's OK.
The Arduino needs at least 6 V input (7 V recommended), but the circuit works at 5 V, and that 5 V is also available on the power connector, at the bottom of the picture.
Connect this 5V and the ground next to it to the relay module's power connector, and up to 8 of the digital I/O's shown at the top to the relay module's logic inputs and you're in business.
edit
vicatcu explains that the Arduino can also be powered from the USB input, and that's true. But the relay module will draw up to 160 mA, and that may be more than your USB port will supply. The Arduino itself also needs around 50 mA.
edit 2012-07-09, re jippie's comment
The Arduino Uno's LDO voltage regulator can supply 800 mA, which should be sufficient for Arduino + relay module, together about 200 mA. At 6 V in the LDO will dissipate 200 mW, and that's no problem, but at 15 V in that becomes 2 W, and that may be too much for the NCP1117's thermal protection. After all this is an SMT device. So it's advisable to use an as low as possible input voltage, or use a separate 5 V wall wart to power the relay module.