Traumatic brain injury such as concussion results when the head undergoes a very large acceleration?

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 1000 m/s2 lasting for at least 1 ms will cause injury. Suppose a small child rolls off a bed that is 0.44 m above the floor. If the floor is hardwood, the child's head is brought to rest in approximately 1.9 mm. If the floor is carpeted, this stopping distance is increased to about 1.2 cm. Calculate the magnitude and duration of the deceleration in both cases, to determine the risk of injury. Assume that 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

hardwood floor magnitude______ m/s2 duration ______ ms

Carpeted floor magnitude_______m/s2 duration ______ ms


so far i got...I need to know how to get hardwood and Carpeted floor magnitude


First, calculate the speed of the infant just before impact:

d=.5*g*t^2
t=sqrt(2*.44/9.81)
=.3 sec

speed = g*t
=9.81*.3
=2.94 m/s

Hardwood

Deceleration distance
d=.5*a*t^2
.0019 =.5*a*t^2
a=.0038/t^2
Average speed
v=.5*a*t
=.5*2.94
=1.47
a=2.94/t

Using the equations together
2.94/t=.0038/t^2
t=.0038/2.94
=1.3 ms
a=_____ m/s^2

Hardwood bad

Carpet

The average speed is still the same
a=2.94/t

the deceleration is
a=.0248/t^2

Again solving simultaneously
t=.0248/2.94
=8.4 ms
a=_____ m/s^2

Carpet= no trauma.

Thank you

Answer 1

Deceleration distance s = at^2/2 = vt/2 = v^2/2a, so
a = v^2/(2s)
= 2275 m/s^2 for hardwood
= 360 m/s^2 for carpet
You could also have gotten about those same answers by continuing:
v = 2.94 m/s
t = 1.3 ms, a = v/t = 2262 m/s^2
t = 8.4 ms, a = v/t = 350 m/s^2