Very hard college physics question! Need help?

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 0.40 m above the floor. If the floor is hardwood, the child's head is brought to rest in approximately 2.2 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

Thanks for any help!

Answer 1

Divide the distance of acceleration by the distance of deceleration and multiply by the acceleration due to gravity:  9.8m/s^2 x .4m/.012m = 817m/s^2 and 9.8m/s^2 x .4m/.0022m = 1782 m/s^2.  Then use the formula t = (2d/a)^1/2  (derived from d = 1/2at^2) to arrive at times of 5.4ms for the rug and 1ms for the hard floor:  (2 x .012m/817m/s^2)^1/2, etc.

Answer 2

I found this question interesting and I didn't know those stats, so thanks for teaching me something.  Let me try to return the favor.

Here's a general way to solve problems: look for what you need, and then figure out how to get that by working backwards.

What do we need here?  Surely the acceleration.  Ok how do we get that?  Well we're given a height, and then a stopping distance.  Well, I know the stopping distance, and if only I knew the velocity upon hitting the floor and the time it took to stop, I could use the simple equation:

acceleration = change in velocity / change in time

To find the deceleration, right?

Ok, so now my goal is to find the change in velocity and the change in time.

The initial velocity with which the head strikes the floor is nothing more than the FINAL velocity, if I consider the problem of: how fast does something hit the ground if dropped at a certain height under the influence of gravity.

Use the equation:  V_final = sqrt(V_initial^2 + 2*A*H)

Where A is the acceleration (9.81, since it's just gravity), and H is the height (.4m).  V_initial is 0, since the child falls from rest.  You can find V_final that way, which is how fast the head is going as it hits the ground.  

Now we know the change in velocity right, because it's just:

Final V - Initial V, where this time (if we're considering when the head hits the ground) the final V is 0.   So the change in velocity is simply equal to the initial velocity (negative because it's stopping).

Now all we need is the find the change in time, and we're golden.  See how we're doing this?  Well for that, we have another distance, and if we have a distance and a rate I can tell you a time by the simple formula D = R * T.

Use that to find the time, which is different in both cases, and you've got your answer.  Just divide the change in velocity (which is the same in both cases) by the separate times (different).

I'll leave it to you to crunch the intermediate steps, but the final answers are:

hard wood: -3566 m/s^2, for .785 ms very serious injury

carpet: -653.8 m/s^2 for 4.3 ms, no injury

Remember to convert all units to meters when solving each equation.

I hope this helps.