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 0.39 m above the floor. If the floor is hardwood, the child's head is brought to rest in approximately 1.8 mm. If the floor is carpeted, this stopping distance is increased to about 1.1 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.

hardwood floor magnitude	m/s2
hardwood floor duration	ms
carpeted floor magnitude    m/s2
carpeted floor duration    ms

Answer 1

Here is what I solved before, please modify the figures as per your question. Please let me know if you have further questions. Ifthis helps then kindly rate 5-stars.

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 0.39 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.
hardwood floor magnitude m/s2
hardwood floor duration ms
carpeted floor magnitude m/s2
carpeted floor duration ms

Let's start by finding the Child's speed when they hit the floor. To find that we can use conservation of energy:

Solving for V gives:

If you don't include the stopping distance that gives a velocity of:

If you don include stopping distance, the speed as the child hits the carpet is:2.72 m/s

and as the child hits the wood: 2.76 m/s (No significant change here)

Now to solve for the acceleration we can use the equation:

and since v_f=0 in both cases:

Here's the final accelerations:

Hitting the wood:

hitting the carpet (not counting carpet thicknes into speed)

hitting the carpet considering carpet thickness:

So, the child is fine on the carpet.

Now we need to find out how long the acceleration is for when the child hits the wood to see if it's long enough to cause injury.

We can use:

and since v_f=0:

So yes, the child would be injured falling onto the wood floor.