Hi just drove my mondeo this morning and after driving not very far the fuel light on the dashboard comes on and it shows empty, If I restart the car it comes on as normal with half a tank left then drops again when driving further

It is a 1.6 2011 diesel with 97000 miles

Will it be safe to drive it like this as the car doesnt stop when this happens and what could be wrong with it?

Possibly the fuel sender in the tank, or poor connection from the tank to the main electrical harness - checking the connection is the easier job, probably under the rear seat there'll be a large round cut-out in the floor carpet. Another potential could be the series of multi-plugs going into the GEM/BCM which probably lives behind the glovebox - check the handbook for location of the cabin fuse box.

If that doesn't help maybe other owners can advise if they've had similar problems.

...I had a vaguely similar problem in a Cougar, but it just went away.

I had parked my car up on what was probably the hottest day of the year, and when I got back to it, three or four hours later, it read zero fuel. First I checked for any smell of petrol anywhere near the car, or puddles under the car, and there wasn't any evidence of fuel actually leaking, so I assume that the car was safe to start.

It started up ok, and as I was within a few hundred yards of a garage, I got some more fuel (so, I'd know that I had fuel, whatever the gauge said).

I went out to look at the problem the following day, and there was a completely normal reading (say, half full) and the problem never came back. I assume that my problem was wiring and something about the weather (temperature) had set it off, but I don't really know. Yours could be wiring, the sender, or the 'receiver' (the module that gets the information).

If you take the wire off the sender, you should be able to measure a voltage (on a DVM). It'll probably be about 12 volts (battery voltage) with the wire to the sender removed, and probably should behave as if there is a 1k Ohm to 5k Ohm resistor in series (for example, if you connect a 1k resistor to ground, the voltage should go down to, say, half the open circuit value). If that's all right then it is either the sender itself (measure resistance!) or the module at the far end is getting the voltage, but not passing on a correct reading to the gauge (wiring to the gauge, gauge itself).

had no problems with it today, maybe it was something to do with the heavy rain

Fuel sender im guessing, a break in the sender circuit at a particular fuel level, so now you used more then its gone lower than the open circuit, refill tank and see if it happens again at simular fuel level.

Sender circuit for sure, it could be the moisture has gotten into an exposed piece of wiring and the cold is affecting the resistance readings going to the ECU and instrument cluster. Alternatively, if the water is getting to the FSU itself, the freezing could be physically affecting the FSU itself.

I filled it up to 3/4 full and have drove around 500 miles and through heavy rain and not had any more problems with it

Excellent well let's just hope the fsu was stuck temporarily!