Question

6. Traumatic brain injury such as concussion results when the head undergoes a very large acceleration. Generally, an acceleration greater than 1000 ms lasting for at least 1 ms will cause traumatic injury. Suppose a small child rolls off a bed that is 0.42 m above the floor. Assume the child remains horizontal during the fall to the floor. If the floor is hardwood, the child's head is brought to rest in a distance of approximately 2.0 mm Note that there are two accelerations to consider in this problem. The first is the acceleration due to gravity as the child falls to the floor. The second is the acceleration as the head comes to rest over a very short distance when hitting the floor. The first acceleration results in an increase in speed while the second acceleration results in a decrease in speed. (a) Calculate the velocity with which the head hits the floor after a fall of 0.42 m. (b) Calculate the magnitude of the acceleration in bringing the child's head to rest by the hardwood. (c) Calculate the time duration of the acceleration in bringing the child's head to rest (d) Is traumatic injury likely to result? Explain Note that a more complicated fall could result in a head velocity greater or less than the speed you calculate.

Answer 1

Initial velocity = 0(time of fall) final velocity = V_1 h = 0.42 m acceleration = g = 9.8 m/s^2 using v^2 - u^2 = 2as v_1^2 - 0 = 2 times g times h V_1 = squareroot 2 times g time h = squareroot 2 times 9.8 times 0.42 = 2.87 m/s b) Now again initial velocity = V_1 final velocity V_2 = 0 v_2^2 - V_1^2 = 2ad 0 - (2.87)^2 = 2 times a times (0.002) d = 2mm = 0.002 m a = -(2.87)^2/2 times (2 times 10^-3) = -2060 m/s^2 so magnitude of a = 2060 m/s^2 c) let time taken = t using v = u + at v_2 = V_1 + at 0 = 2.87 - 2060t t = 2.87/2060 = 1.39 times 10^-3 sec = 1.39 ms d) Yes, as a = 2060 m/s^2 and t = 1.39 ms a > 1000 m/s^2 t > 1 ms