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If I am flying a multirotor drone and it runs out of battery mid-air, what will happen to it? Will it just fall down from the sky or will it give me a low-battery warning?

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Assuming that you ignore all of the warning signs that your battery is low, while also assuming that you can completely deplete the voltage in your battery, the drone will ultimately reach a point where it is still providing power for the motors to spin the props, but the props won't provide enough lift to keep the drone in the air.

The drone will start to descend slowly and will increase descent the longer time goes on. If you are a few feet off the ground, it might look like a soft landing. If you are a few hundred feet off the ground, it might be a hard crash. There will also be a point where the voltage cannot provide enough power for the flight controller and receiver and other electronics on the drone, which will shut them down and at that point, if the drone is still able to fly, it will drop since motors will not be spinning at this point.

The last scenario is not likely to happen since the main electronics tend to run at very low voltages, so they will have power for a while after the motors stop spinning fast enough to provide lift to the drone.

The drone won't just give you a warning when the battery is low. You, or the maker of the drone, need to set that up. If it's already set up, then you should have all you need to avoid a low voltage scenario.

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It really depends on the drone.

Photography quads like those build by DJI will try to return to home. If you cancel RTH, they will try to softly land themselves.

FPV quads will try to warn you through a beeper, telemetry and OSD, but if you don’t have any GPS set up they will just land (fairly hard) wherever they are. It also depends on voltage sag. If you’re just cruising, it won’t sag too much and will land more softly, Because there's just not enough power left in the battery to keep or gain the altitude but enough to keep the motors spinning. If you’re catching yourself after a power loop, there’ll be a large voltage sag and you’ll land harder (crash).

You grade quads tend to have their lights start to flash, and then will slowly fall down.

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This depends on the drone. Most drones have a low voltage cutoff, meaning it will notify you of low battery and you can start descending or do a forced landing. You will have no power but you will still have control of the drone. Only badly built drones will just fall to the ground with no indication what-so-ever.

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There are a range of possibilities, depending on the make and type of drone. You should check your drones manual for details.

Most of the multirotors I have flown have been DJI aircraft, and these give an audible, visual and haptic alarm to the pilot when the battery passes set 'low' and 'critical' levels.
If the battery reaches the point where the aircraft can only just return home, the aircraft will automatically begin it's RTH procedure; this can be overridden manually but may not be a good idea.
If the battery reaches the point where it cannot continue flight for much longer (this varies, depending on factors like current altitude), the aircraft will descend where it is in a controlled manner and land.

I also have a small "Micro Drone" which I can fly indoors. On this drone, the LEDs flash on low battery but also as the charge decreases so does the thrust. While the motors keep spinning, eventually it just can't generate enough lift to maintain altitude and will slowly descend to the ground. (I tested this one with a low hover, to see what would happen.)

On the fixed wing aircraft I fly, the reducing battery results in reduced thrust. This results in the aircraft naturally starting to descend and as the motor needs more power than the servos you can typically retain control of the aircraft while it glides back to the surface. Landing without the motor is a slightly different experience, so fixed-wing pilots will regularly practice power-off landings.

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