# Distance and Speed

When I look up in the sky and I see a plane moving and it's further away why does it seem to move so slow compared if a flock of geese flies over at same time and seems to travel faster?

What you are measuring is the angular speed $\dot{\theta}$ of the plane and the geese. The speed of the plane or geese with respect to the ground will be $L \times (\cos \theta) \dot{\theta}$ where $L$ is their distance (if you look at the triangle formed by the solid lines some simple trigonometry gives $d = L \sin \theta$). The important thing is that an object further away from you covers more (horizontal) distance than for a nearer object to you when it moves through the same range of angle (you can see that the dotted line is shorter in length).

• $L\dot{\theta}$ isn't quite right... – lemon May 21 '16 at 20:48

Artists learn to draw using perspective - the further away an object, the smaller it appears.

The series of identical boxes are pictured above, with perspective used to show their apparent size from the artists eye. A runner passing one box per second would appear small in the distance, but would be running at a constant speed.

The situation described in the OP lacks the background of boxes, but can be evaluated similarly: the geese are large birds, but very small when compared to an airliner, so if the flock of geese will be much closer than the airliner.

If you now imagine two rows of boxes strung across the sky, one near the flock of geese, and the second, distant row, near the airliner, you will have a scale to measure their respective speeds.

In your mind's eye, you will realize that the geese are flying much, much slower, usually 5% of the speed of an airliner.