The correct answer is option 3 - Doppler effect.

The Doppler effect is a phenomenon that occurs when there is relative motion between a source of waves (in this case, the radar gun) and the receiver of those waves (in this case, the police car). The radar gun emits radio waves which bounce off the moving vehicle and return back to the radar gun.

As the vehicle is moving towards the radar gun, the waves are compressed, resulting in a higher frequency. On the other hand, when the vehicle is moving away from the radar gun, the waves are stretched out, leading to a lower frequency.

By measuring the change in frequency of the returned waves, the radar gun can calculate the speed of the vehicle. This is because the frequency shift is directly proportional to the speed of the vehicle.

So, in summary, the radar gun used by the police to check over-speeding vehicles works by utilizing the Doppler effect to measure the change in frequency of the waves reflected off the moving vehicle.