Speed of a fly inside a car


Basically from the frame of observation as your car:

The fly was inside your car, so its speed with respect to the car is zero. Its just as much inside the car as you are. Both are travelling at 120 with respect to any observer on the road. ut with respect to anyone inside car you both are just sitting inside the car.

So the speed of fly with respect to you is $v=0\,\frac{m}{s}$, with respect to some observer on the road is $120\,$km/h.

Its no more than a tissue paper you might keep near the steering wheel, in front of you.

For the same reason that when you are in a train that moves at 120 km/h, you can walk down the aisle without running at 120 km/h yourself.

In your case, you are walking on a floor that is moving at 120 km/h alredy. In the case of a fly, it is flying in air that is moving at 120 km/h.

Well, probably one of the first person to note and describe that was Galileo. As he beautifully describes in his Dialogue Concerning the Two Chief World Systems:

Shut yourself up with some friend in the main cabin below decks on some large ship, and have with you there some flies, butterflies, and other small flying animals.

With the ship standing still, observe carefully how the little animals fly with equal speed to all sides of the cabin. When you have observed all these things carefully (though doubtless when the ship is standing still everything must happen in this way), have the ship proceed with any speed you like, so long as the motion is uniform and not fluctuating this way and that. You will discover not the least change in all the effects named, nor could you tell from any of them whether the ship was moving or standing still.

Finally the butterflies and flies will continue their flights indifferently toward every side, nor will it ever happen that they are concentrated toward the stern, as if tired out from keeping up with the course of the ship, from which they will have been separated during long intervals by keeping themselves in the air.

This observation lead Galileo to enunciate that the laws of mechanics are invariant under a change of inertial frames. Nowadays we believe that not only the laws of mechanics but also the laws of physics shall be invariant under the change of any frame of reference. This is a fundamental principle of physics, called the Relativity Principle.

Restricting ourselves to mechanics, it means that there is no mechanical experiment capable of detecting the absolute motion of an inertial frame. This is actually contained in the first Newton's law (which is due to Galileo). When the fly rests on the car it has $120\, km/h$ forwards with respect to the ground. When it jumps upwards, the first law asserts that it must keep this $120\, km/h$ forwards since there is no force pushing or pulling it in the horizontal direction. It can fly forward at $5\, m/s$ respective to the car exactly in the same way it would fly respective to the ground if it was outside the car.