Tuesday, December 3, 2013

Can I jump from a tree on willow path without breaking my leg?

Can I jump from a tree on willow path without breaking my leg?

The average height of a willow tree is 40ft (.304m/1ft) = 12.16m
I am assessing whether my tibia will break. The average diameter of a tibia is 2.5cm; r = .0125m

The compressive strength of bone = 170x106 N/m2
F/A = compressive strength
Thus the max Force from the ground that the bone can withstand = A x compressive strength
Max Fground = (πr2)(compressive strength) = (π * .0125m2)( 170x106 N/m2) = 83448.6N

Use W and Energy principles in order to obtain the Force exerted by the ground on the bone.
W = FdcosΘ where d is the distance over which the Force of the ground acts to stop my jump. I will assume that this distance = .50m
Since the Force of the ground acts on my leg in the direction opposite my motion (we are assuming that I jump vertically from the tree) Θ = 180°
Thus W = -Fd = -F(.50m)

From the Work-Energy principle we have:
Wnet = ∆KE = -∆PE + Wnc
There is no Wnc, and it is easiest to just look at ∆PE
Wnet = PEf – PEo = 0 – mgh where h is the height of the tree

Now we can put the equations together:
-mgh = -Fd
F = mgh/d
I am taking my mass to be approximately 50kg
F = (50kg)(9.8m/s2)(12.16m)/0.50m = 11916.8N

Thus I can jump from the tree without breaking my tibia!
However, you can see from the derived equation that there is an increased risk of breaking your leg the higher the place you jump from and the greater your mass.