A few weeks ago, I got elbowed in the head during basketball practice and was diagnosed with a concussion. I began to wonder about the physics of what I had experienced. How much force was applied to my head from the elbow? How fast was the elbow going when it hit my head? With some research and some approximations I was able to answer these questions.
A concussion is a traumatic brain injury that is caused from a collision or whiplash where the brain and the head shake rapidly back and forth. Symptoms are highly variable depending upon the severity of the injury, but may include memory problems, confusion, irritability, balance problems, and dizziness. Recovery also varies with severity, but may last several days to several months. Most concussions deliver about 95 g's upon impact.
The upper arm of an adult male is about 3.25% of the body mass. If the person who elbowed me were 97 kg, then the mass of the elbow would be 3.2 kg. With an elbow of this mass, 1 g would be equivalent to 31 N (mass x gravity). Thus, the force necessary for the elbow to cause a concussion would be 2945 N (95 g's). With the force that was applied to my head known, Newton's second law can be applied to find the speed at which I was hit.
F=ma
If the force was 2945 N, and the mass was 3.2 kg, then the acceleration would have been 920 m/s^2. The elbow was at rest about arm's length away before it collided with my head. My arm is 0.84 meters long. With a simple kinematic equation, the speed of the elbow can be found.
vf^2 = vo^2 + 2a(xf − xo )
By using this equation and solving for the final velocity, the velocity of the elbow was found to be 39 m/s. 39 m/s is equal to 87 mph.
This research found that my concussion was caused by an elbow applying 2945 N to my head at a speed of 87 mph.
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