Question

Traumatic brain injury such as concussion results when the head undergoes a very large acceleration. Generally, an acceleration less than 800 m/s2 lasting for any length of time will not cause injury, whereas an acceleration greater than 1,000 m/s2 lasting for at least 1 ms will cause injury. Suppose a small child rolls off a bed that is 0.37 m above the floor. If the floor is hardwood, the child's head is brought to rest in approximately 2.1 mm. If the floor is carpeted, this stopping distance is increased to about 1.4 cm. Calculate the magnitude and duration of the deceleration in both cases, to determine the risk of injury. Assume the child remains horizontal during the fall to the floor. Note that a more complicated fall could result in a head velocity greater or less than the speed you calculate.

Answer 1

Answer:

1728.42857143 m/s²

0.00155883061577 s

259.264285715 m/s²

0.0103922041051 s

The child will get injured if he/she falls on a hardwood floor

Explanation:

t = Time taken

u = Initial velocity

v = Final velocity

s = Displacement

a = Acceleration

g = Acceleration due to gravity = 9.81 m/s²

$v^2-u^2=2gs\\\Rightarrow v=\sqrt{2gs+u^2}\\\Rightarrow v=\sqrt{2\times 9.81\times 0.37+0^2}\\\Rightarrow v=2.69432737432\ m/s$

$v^2-u^2=2as\\\Rightarrow a=\dfrac{v^2-u^2}{2s}\\\Rightarrow a=\dfrac{0^2-2.69432737432^2}{2\times 2.1\times 10^{-3}}\\\Rightarrow a=-1728.42857143\ m/s^2$

Magnitude of deceleration is 1728.42857143 m/s²

$v=u+at\\\Rightarrow t=\dfrac{v-u}{a}\\\Rightarrow t=\dfrac{0-2.69432737432}{-1728.42857143}\\\Rightarrow t=0.00155883061577\ s$

Time taken is 0.00155883061577 s

$v^2-u^2=2as\\\Rightarrow a=\dfrac{v^2-u^2}{2s}\\\Rightarrow a=\dfrac{0^2-2.69432737432^2}{2\times 1.4\times 10^{-2}}\\\Rightarrow a=-259.264285715\ m/s^2$

Magnitude of deceleration is 259.264285715 m/s²

$v=u+at\\\Rightarrow t=\dfrac{v-u}{a}\\\Rightarrow t=\dfrac{0-2.69432737432}{-259.264285715}\\\Rightarrow t=0.0103922041051\ s$

Time taken is 0.0103922041051 s

It is likely that the child will get injured if he/she falls on a hardwood floor.

It is less likely that the child will get injured if he/she falls on a carpeted floor.