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 0.39 m above the floor. If the floor is hardwood,the child's head is brought to rest in approximately 2.2 mm. If the floor is carpeted, this stoppingdistance is increased to about 1.1 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

Since V=at then t= V/a And since h=0.5at2 Now we have h=0.5a(V/a)2 Finally h=0.5V2 /a and now we have an expression for acceleration a in terms of speedof impact V and the travel distance after the impact h. a=0.5V2 /h . To find speed we can use potential and kinetic energy Pe(top) = Ke(bottom) mgH = 0.5mV2 where H= 0.39m V2= 2gH _________________ to find acceleration a=0.5V2 /h and now a= gH /h a(hardwood) = gH /h = 9.81 x 0.39/ 0.0022 = 1740m/s2a(carpeted) = gH /h =9.81 x 0.39/ 0.011 = 348 m/s2

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