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I was riding a bus one day and noticed that the double windows had some water between them. As the bus accelerated, the water collected to the sides, first forming a trapezoid and then a right triangle.

I begun wondering how it would be possible to measure the acceleration using only the geometry of the form of the amount of water.

Like, assume that at $a=0$ the water has height $h$ and width $w$. As the bus accelerates, at some time $t_1$ the water forms a trapezoid with the shorter side $h_1$ ans longer side $h_2$. The bottom has same width $w$.

At time $t_2$ water has formed a triangle with height $H$ and width $w$ (the last moment it touches the other side of the glass). And finally at time $t_3$ the height is $y$ and the width is $W$.

In each situation, what is the acceleration of the bus?

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I think the answer can be relatively simple. The acceleration of the bus is perpendicular to the direction of gravity. Assuming that the water-air interface stays flat, you can approximate the tilt angle of the interface with respect to the horizontal, say $\alpha$, and calculate the acceleration by $a=g\tan\alpha$.

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    $\begingroup$ This assumes, of course, that the water is not sloshing about, i.e., that the response time of the water is quicker than any changes in acceleration. (Otherwise, there's nothing to do.) $\endgroup$ – Emilio Pisanty Feb 23 '13 at 23:39
  • $\begingroup$ Oh, that simple. Thanks :D And of course I assume that the water flows smoothly. In reality it is quite messy. $\endgroup$ – Valtteri Feb 23 '13 at 23:43
  • $\begingroup$ To add to this (correct) answer, this works because inside of the bus, the effective acceleration due to gravity is the sum of the (fictitious) acceleration (pointing backward equal to the magnitude of the bus's acceleration), and the acceleration due to gravity. In addition, the surface of the water will be perpendicular to the direction of the effective acceleration due to gravity in the bus. $\endgroup$ – joshphysics Feb 24 '13 at 1:16
  • $\begingroup$ + nice answer. There's a simple way to measure it. Measure the height $h$ and base $b$ of the triangle. Then the acceleration of the bus is $g$ times $h/b$ (assuming driving on a level street). $\endgroup$ – Mike Dunlavey Feb 24 '13 at 21:20

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