Help me solve this!!!

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 0.42 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 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.

A.) What is the hardwood floor magnitude in m/s²?

B.) What is the carpeted floor magnitude in m/s²?

Answer 1

This is the general strategy:

(I) First calculate the child's velocity v at point of impact,

(II) from than calculate deceleration given the stopping distance.

Both calculations use these two equations (which come from the definition of acceleration):

v = at (1)

s = at²/2 (2)

where

a is acceleration

t is time it takes to change velocity from 0 to v, or from v to 0

s is the distance travelled during the time t.

Let's prepare the following algebraic manipulations.

From (2) t = √(2s/a) (3)

From (1) and (3) v = a√(2s/a) = √(2sa) (4)

From (4) a = v²/2s (5)

Let

s₁ = 0.42 m be the distance of the fall,

s₂ = 1.9 mm or 1.2 cm be the stopping distance,

t₁ (unknown) be the time it takes to fall,

t₂ (unknown) be the duration of deceleration,

v (unknown) be the velocity at impact,

a₁ = 9.81 m/s² be gravitational acceleration (experienced during fall)

a₂ (unknown) be the deceleration experience during impact.

(I)

To calculate v, use equation (4) substituting s = s₁, a = a₁

Then

v = √(2s₁a₁) (6)

(II)

To calculate a₂ and t₂, substituting s = s₂, a = a₂, and v from (6).

Then using (5) and (6)

a₂ = v²/2s₂ = 2s₁a₁/2s₂ = (s₁/s₂)a₁

Using (3)

t₂ = √(2s₂/a₂) = √(2s₂/((s₁/s₂)a₁) = √(2s₂²/(s₁a₁) = s₂√(2/(s₁a₁))

Now substitute actual numbers:

The case of hardwood

a₂ = (0.42/0.0019)×9.81 = 2,168 m/s²

t₂ = 0.0019×√(2/(0.42×9.81) = 0.0013 s = 1.3 ms

The case of carpet

a₂ = (0.42/0.021)×9.81 = 196 m/s²

t₂ = 0.021×√(2/(0.42×9.81) = 0.015 s = 15 ms