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## 'How Do You React?' printed from http://nrich.maths.org/

Herschel of the European School of Varese
sent us a very succinct solution.

Speed-time graph:

Since gravity exerts a constant force, the acceleration is also
constant: $9.8ms^{-2}$. Therefore, speed is proportional to time
and the speed-time graph shows a straight line (with a gradient of
$9.8$).

Distance-time graph:

The distance travelled depends on the speed, which increases over
time. Therefore, the distance travelled in the first second is less
than that travelled during the second, which is in turn less than
that travelled during the third. The graph is therefore a
half-parabola with the gradient increasing over time. This means
that it takes longer for the ruler to travel the first $15cm$
compared to the next $15 cm$, and a reading of $30 cm$ therefore
indicates a reaction time that is slower, but not twice as slow!

To look at it algebraically, distance is calculated by the formula
$d=\frac{1}{2}\times a\times t^2$. This means that the distance is
NOT proportional to the reaction time, but instead to the SQUARE of
the reaction time. Double the distance therefore means a reaction
time that is 1.41 (i.e. the square-root of $2$) times slower rather
than twice as slow. Rearranging the formula above to make $t$ the
subject gives us $t=\sqrt{\frac{2d}{a}}$, which means the reaction
time for $15cm$ is $\sqrt{0.03}=0.17$ seconds, while that for
$30cm$ is $\sqrt{0.06}=0.24$ seconds - only $0.07$ seconds slower,
or $41$%.

As for the "typical" $0.2$ seconds reaction time, the typical
distance should be $0.5\times 9.8\times0.2^2=0.196m$ which is
$19.6cm$. I found it hard to get consistant results for my own
reactions - my distances ranged from $15cm$ ($0.17$ seconds) to
$25cm$ ($0.23$ seconds). Incidentally, it's funny to note that a
distance of $0.204$ meters ($20.4$ centimeters) corresponds
precisely to a time of $0.204$ seconds!