Traumatic brain injury such as concussion results when the head undergoes a very

Question

Traumatic brain injury such as concussion results when the head undergoes a very large acceleration. Generally, an acceleration less than 800 m/s2 lasting for any length of time will not cause injury, whereas an acceleration greater than 1000 m/s2 lasting for at least 1 ms will cause injury. Suppose a small child rolls off a bed that is 0.39 m above the floor. If the floor is hardwood, the child's head is brought to rest in approximately 2.1 mm. If the floor is carpeted, this stopping distance is increased to about 1.0 cm. Calculate the magnitude and duration of the deceleration in both cases, to determine the risk of injury. Assume the child remains horizontal during the fall to the floor. Note that a more complicated fall could result in a head velocity greater or less than the speed you calculate.

hardwood floor magnitude     m/s2 hardwood floor duration     ms carpeted floor magnitude     m/s2 carpeted floor duration     ms

Explanation / Answer

speed of the child after falling a height h , v = sqrt(2*g*h)

v = sqrt(2*0.39*9.8)

v = 2.76 m/s

on hardwood floor

initial speed vi = v = 2.76 m/s

final speed vf = 0

distance fallen y = 2.1 mm = 0.0021 m

from equatio of motion

vf^2 - vi^2 = 2*ay*y

0 - 2.76^2 = 2*ay*0.0021

deceleration = 1813.7 m/s^2            <<<---------------answer

from equation y = (vi + vf)*t/2

t = 2*y/(vi + vf)

t = 2*0.0021/(2.76+0)

t = 0.0015 s = 1.5 ms                  <<<---------------answer

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on carpeted floor

initial speed vi = v = 2.76 m/s

final speed vf = 0

distance fallen y = 1 cm = 0.01 m

from equatio of motion

vf^2 - vi^2 = 2*ay*y

0 - 2.76^2 = 2*ay*0.01

deceleration = 380.88 m/s^2   <<<---------------answer

from equation y = (vi + vf)*t/2

t = 2*y/(vi + vf)

t = 2*0.01/(2.76+0)

t = 0.0072 s = 7.2 ms        <<<---------------answer

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