Traumatic brain injury such as concussion results when the headundergoes a very

Question

Traumatic brain injury such as concussion results when the headundergoes a very large acceleration. Generally, an accelerationless than 800 m/s2 lasting for any length of time willnot cause injury, whereas an acceleration greater than 1000m/s2 lasting for at least 1 ms will cause injury.Suppose a small child rolls off a bed that is 0.42 m above the floor. If the floor is hardwood, thechild's head is brought to rest in approximately 2.2 mm. If the floor is carpeted, this stoppingdistance is increased to about 1.0 cm.Calculate the magnitude and duration of the deceleration in bothcases, to determine the risk of injury. Assume that the childremains horizontal during the fall to the floor. Note that a morecomplicated fall could result in a head velocity greater or lessthan the speed you calculate.

Explanation / Answer

   In this question you have to first of all findout the velocity / speed attained by the child's head when it hitsthe floor.    u = 2gh = (2*9.8*0.42)m/s    Now, this velocity/speed becomes zero at rest incovering d=0.0022 m    So, v2 = u2 - 2ad    => 0 = 2gh - 2*a*0.0022    => a = 1870.91 m/s2     Now, in the case of the carpetedfloor,     2gh = 2a'*0.01       => a' = 411.6 m/s2    Hence, we get the decelerations in bothcases.    To calculate the periods of deceleration,    use the equation, v = u - at v = u - a't' In both the cases v at rest = 0 => t = u/a, t' = u/a'    u = 2gh = 2.869 m/s    you know a and a' from above.    So, periods of retardation are :    t = 1.533 ms    And, t' = 6.97 ms Clearly the first case will result in injury while thesecond case would not.

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