# Watching an aeroplane through a window

#### bill nye scienceguy!

As I was sitting at work the other day productively watching an aeroplane fly across the window I started to think of this problem.

If I measure the time it takes for the image of the plane to track across a known distance on the window I can quickly calculate a speed at which the image is travelling. Assuming I know the distance of the plane, how can I extrapolate its absolute velocity?

Is it as simple as using similar triangles to estimate the absolute distance that the plane has travelled in the measured time and calculating from that or do I need to get reference frames involved?

Obviously this approach assumes I can accurately estimate how far away a plane is...

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#### Borek

Mentor
Triangles should be enough.

#### Hassan2

If the flight direction is not parallel with the window, things may go a little more complicated.

#### bill nye scienceguy!

I was assuming that the travel was parallel to the window, in which case my triangles method is good enough.

If it were going into or away from the window, how would I go about it? I used to be pretty hot at transforming between different coordinate systems in 2d solid mechanics but dynamics was never my strong suit.

#### Borek

Mentor
Still a triangle, just you will need two distances - one for each side of the window.

#### Hassan2

If you have the following data:

W: window width,
h: the distance from the viewer to the window,
β: the angle between the flight direction and the window ( positive angle when it is getting away from and negative when its coming toward the window)

t: the measured time
L: the distance from the viewer to the plane when it first appears in the window frame,

then for the speed v, you would come up with the following expression:

v=$\frac{4hWL}{t\sqrt{W^{2}+4h^{2}}}$$\frac{1}{4hcosβ-Wsinβ}$

It can be seen that for some large angles, v becomes infinite. This is because the plane doesn't disappear from the one sides of the window.

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