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

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 0.36 m above the floor. If the floor is hardwood, the child's head is brought to rest in approximately 2.3 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.

I was able to find the magnitude for both, but the durations keep coming out wrong. These are the anwers I keep getting:

1533.913043 m/s^2 = hardwood floor magnitude

0.001731723 m/s = hardwood floor duration

252 m/2^2 = carpet floor magnitude

0.010540926 m/s = carpet floor duration

Explanation / Answer

here,

height of the bed , h = 0.36 m

velocity gained by child before hiting the ground , v = sqrt(2*g*h )

v = 2.66 m/s

for the floor of hardwood,

stoping distance , d' = 2.3mm = 2.3 * 10^-3 m

let the deaccelration be a

0 - 2.66^2 = 2*a * 2.3 * 10^-3

a = 1538.17 m/s^2

v = u + a*t

0 = 2.66 - 1538.17 * t

t = 1.73 *10^-3 s

t = 1.73 ms

so the time is alos greater than 1 ms and the accelration is also greater than 1000 m/s^2 ,

so the hardwood cause a injury

b)

for the floor of carpet,

stoping distance , d' = 1.4 cm = 1.4 * 10^-2 m

let the deaccelration be a

0 - 2.66^2 = 2*a * 1.4 * 10^-2

a = 252.7 m/s^2

v = u + a*t

0 = 2.66 - 252.7 * t

t = 10.53 *10^-3 s

t = 10.53 ms

the accelration is less than 800 m/s^2 and duration is 10.53 ms

so no injury is caused

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