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 0.55 m above the floor. If the floor is hardwood, the child's head is brought to rest in approximately 1.7 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

a) wooden floor

a = 3170.6 m / s², t = 1.03 10⁻³ s

the child's traumatic injury to the brain

b) the floor is carpeted

a = 385 m / s²

no injuries are created in the child

Step-by-step explanation:

To solve this exercise we can use the energy conservation relations, let's start by looking for the speed of the child when he is just reaching the ground

starting point. When you get out of bed

Em₀ = U = m g h

final point. Just when it hits the floor

`Em_(f)` = K = ½ m v²

as there is no friction, energy is conserved

Em₀ = Em_{f}

mgh = ½ m v²

v² = 2 gh

let's calculate

v² = 2 9.8 0.55

v² = 10.78

v = 3.28 m / s

Now let's use the concepts of kinematics to find the deceleration. The case of the wooden floor, where the distance for the deceleration is

x = 1.7 mm = 0.0017 m

v² = v₀² - 2 a y

as the child stops the final velocity is zero

0 = v₀² - 2a y

a = v₀² / 2y

let's calculate

a =
`(10.78)/(2 \ 0.0017)`

a = 3170.6 m / s²

Let's find the time that braking lasts

v = v₀ - a t

0 = v₀ - at

t = v₀ / a

t = 3.28 / 3170.6

t = 1.03 10⁻³ s

hence the child's traumatic injury to the brain

second case the floor is carpeted, in this case the stopping distance is

x = 1.4 cm = 0.014 m

we look for acceleration

a = v₀² / 2y

a =
`(10.78)/(2 \ 0.014)`

a = 385 m / s²

therefore no injuries are created in the child

In conclusion we see that with the wooden floor there is silence and with the carpeted floor there is no