Please help me with Physics!?

Question

Here is the problem that I am having a horrible time with:

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 0.40 m above the floor. If the floor is hardwood, the child's head is brought to rest in approximately 2.0 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

Answer 1

OK, to solve this problem, you need to break it into three parts.

Step 1: Find the velocity of the child when they hit the ground.
You can use Torricelli's equation to determine velocity independent of time: vf^2 = vi^2 + 2ad
where: vf = final velocity, vi = initial velocity.

vf^2 = 0 + 2 * 9.8 m/s^2 * .4m
vf^2 = 7.84 m^2/s^2
vf = 2.8 m/s

Step 2: Find the Hardwood Floor magnitude and duration.
Use Torricelli's equation again:
vf^2 = vi^2 + 2ad
d = 2 mm = .002 m

0 = (2.8 m/s)^2 + 2 * a * .002m
0 = 7.84 m^2/s^2 + .004a m
-.004a m= 7.84 m^2/s^2
a = 7.84 m^2/s^2 / -.004m
a = -1960 m/s^2 (Solution! Magnitude = 1960 m/s^2)

To find duration, use vf = vi + at.
0 = 2.8 m/s + -1960 * t
-2.8 = -1960 t
.0014285s = t
t = 1.43 ms (Solution, Duration)

Step 3: Find the Carpeted Floor magnitude and duration.
Use Torricelli's equation again:
vf^2 = vi^2 + 2ad
d = 1.2 cm = .012 m

0 = (2.8 m/s)^2 + 2 * a * .012m
0 = 7.84 m^2/s^2 + .024a m
-.024a m= 7.84 m^2/s^2
a = 7.84 m^2/s^2 / -.024m
a = -326.666... m/s^2 (Solution! Magnitude = 327 m/s^2)

To find duration, use vf = vi + at.
0 = 2.8 m/s + -326.666... * t
-2.8 = -326.666... t
.0085714s = t
t = 8.57 ms (Solution, Duration)