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 1,000 m/s2 lasting for at least 1 ms will cause injury. Suppose a small child rolls off a bed that is 0.63 m above the floor. If the floor is hardwood, the child's head is brought to rest in approximately 2.3 mm. If the floor is carpeted, this stopping distance is increased to about 1.4 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

Take the case of hardwood floor -

Velocity after 0.63 m. fall = sqrt [2gh] = sqrt[2*9.81*0.63] = 3.52 m/s

Acceleration on hardwood landing = (v^2/2d) = 3.52^2/0.0023m = 5387 m/sec^2.
Duration of acceleration = (v/a) = 3.52 / 5387 = 6.534 x 10^-4 s = 0.6534 ms.

Now consider the case of carpeted floor -

Acceleration on carpet landing = (v^2/2d) = 3.52^2/0.014 = 885 m/sec^2.
Duration of acceleration = (v/a) = 3.52 / 885 = 0.003977 s = 3.977 ms

This approximates to 4.0 ms.

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