Estimate the time required for the sound to travel a distance of 10 m, be detected by the ear, travel as a nerve impulse to the brain, be processed, then travel as a nerve signal for muscle action back to an arm, and finally cause a muscle contraction. Nerve impulses travel at a speed of about 120 m/s in humans. Distances traveled by the nerve impulses are 10 cm to the brain and then 75 cm from the brain to the arm. Assume that the detection time of the ear and the reaction time of the muscle are about 50 ms and that the brain processing time adds a further 100 ms.

- First of all, let's calculate the time the sound takes to travel through air a distance of d=10 m. The speed of sound in air is v = 343 m/s, so we have

[tex]t_1=\frac{d}{v}=\frac{10 m}{343 m/s}=0.029 s[/tex]

- Then, we have to add the detection time of the ear:

[tex]t_2 = 50 ms = 0.050 s[/tex]

- Then, the time the nerve impulse takes to travel from the ear to the brain (speed v=120 m/s, distance d=10 cm=0.01 m):

[tex]t_3 = \frac{d}{v}=\frac{0.01 m}{120 m/s}=8.3\cdot 10^{-5} s[/tex]

- Then, the brain processing time:

[tex]t_4 = 100 ms = 0.1 s[/tex]

- Then, the time the nerve impulse takes to travel from the brain to the muscles (speed v=120 m/s, distance d=75 cm=0.75 m):

[tex]t_5 = \frac{d}{v}=\frac{0.75 m}{120 m/s}=0.00625 s[/tex]

- Finally, the reaction time of the muscles:

[tex]t_6 = 50 ms = 0.05 s[/tex]

Summing all together:

[tex]t=0.029 s +0.050 s +8.3\cdot 10^{-5} s+0.1 s+0.00625 s+0.05 s=0.235 s = 235 ms[/tex]