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jessica1071 · Feb 19, 2010, 10:57 AM

In a rear–end collision the automobile that is hit is accelerated to a velocity v in 0.01 s. What is the minimum velocity at which there is danger of neck fracture from whiplash? Use the fact that the stretch rupture strength or maximum pressure that a bone can withstand without fracture is 83 MPa, and assume that the area of the cervical vertebra is 1 square centimeter and the mass of the head is 5 kg.

ebaines · Feb 19, 2010, 11:17 AM

Not enough information is given to solve this problem. We need to know:

A. The distance of the center of mass of the head to the vertebra - needed to calculate the moment (torque) being applied to the vertebra:

$ M = m*a*L $

where $ a = \frac {\Delta v} {\Delta t} $

B. The cross-section shape of the vertebra - needed to determine how this torque is translated into stress. You tell us its area is 1 cm^2, but not what the cross-section shape is, and hence we don't know the vertebra's moment of inertia (I) nor it's half height(c). If we have this information then stress can be calcualted from:

$ \sigma = \frac {Mc} I $