The elastic energy stored in your tendons can contribute up to 35 \% of your energy needs when running. Sports scientists have studied the change in length of the knee extensor tendon in sprinters and nonathletes. They find (on average) that the sprinters' tendons stretch 41 {\rm mm}, while nonathletes' stretch only 33 {\rm mm}.The spring constant for the tendon is the same for both groups, 31 {\rm {N}/{mm}} . What is the difference in maximum stored energy between the sprinters and the nonathletes?Express your answer using two significant figures in J.

In such a case where the energy stored is in the physical dimension of the material due to the elastic deformation is called the elastic potential energy.

It is mathematically given as:

[tex]U=\frac{1}{2} k.\Delta x^2[/tex].....................................[1]

where :

U= elastic potential energy

k= constant of elasticity or the spring constant

[tex]\Delta x[/tex]= change in the length

Given:

stretch in tendons for the sprinters, [tex]\Delta x_s[/tex]= 41 mm

stretch in tendons for the nonsprinters, [tex]\Delta x_n[/tex]= 33 mm

spring constant for the tendons of both group, k= 31 [tex]N.mm^{-1}[/tex]

So, the difference in the stored energy:

[tex]\Delta U=[\frac{1}{2} k. x_s^2]-[\frac{1}{2} k. x_n^2][/tex]

[tex]\Delta U=\frac{1}{2} k[ x_s^2-x_n^2][/tex]

putting the respective values

[tex]\Delta U=\frac{1}{2}\times 31 \times [ 41 ^2-33^2][/tex]

[tex]\Delta U=9176\, N.mm[/tex]

[tex]\Delta U=9.176\, J[/tex]