91Ó°ÊÓ

The given data can be listed below,

• The mass of Tarzan is,$m=100\mathrm{kg}$ .
• The initial speed of Tarzan at rest is,u=0
• The height of dangling feet above the ground is,$h_1=2.1\mathrm{m}$ .
• The height of the center of mass above ground is,$h=2.9\mathrm{m}$ .
• The height of the center of mass on the ground is,$h_{2}2.9-2.1=0.8\mathrm{m}$ .

The crouched position of the above-ground is,$h_3=0.5\mathrm{m}$ .

The equation of motion is defined as the simple formula that calculates the displacement, velocity (initial or final), and acceleration of the object relative to the frame of reference.

Using $3^{rd}$equation of motion is given as follows,

$v^2=u^2+2g{h}_1$

Here u is the initial speed, v is the final speed, and g is the acceleration due to gravity.

Substitute 0 for u,$9.8\mathrm{m}/{\mathrm{s}}^2$ for g, and 2.1m forrole="math" localid="1656908484636" $h_1$ in the above equation.

$$\begin{gathered} v^2=0+2\left(9.8\mathrm{m}/{\mathrm{s}}^2\right)\left(2.1\mathrm{m}\right) \\ \mathrm{v}=\sqrt{41.16} \\ \mathrm{v}=6.42\mathrm{m}/\mathrm{s} \end{gathered}$$

Thus, the speed at the instant just before Tarzan's feet touch the ground is 6.42m/s .

The initial internal energy is given as,

$$\begin{gathered} U_1=PE+KE \\ =mg{h}_2+\frac{1}{2}m{v}^2 \end{gathered}$$

Substitute 100kg for m,$9.8m/s^2$ for g,0.8m for$h_2$ and 6.42m/s for v in the above equation.

$$\begin{gathered} U_1=\left[\left(100\mathrm{kg}\right)\left(9.8\mathrm{m}/{\mathrm{s}}^2\right)\left(0.8\mathrm{m}\right)+\frac{1}{2}\left(100\mathrm{kg}\right){\left(6.42\mathrm{m}/\mathrm{s}\right)}^2\right]\left(\frac{1\mathrm{J}}{1\mathrm{kg}.{\mathrm{m}}^2/{\mathrm{s}}^2}\right) \\ =2844.82\mathrm{J} \end{gathered}$$

Now, the final internal energy is given as,

$U_2=mg{h}_3$

Substitute 100 kg for m,$9.8m/s^2$ for g, and 0.5m for$h_3$ in the above equation.

$$\begin{gathered} U_2=\left(100\mathrm{kg}\right)\left(9.8\mathrm{m}/{\mathrm{s}}^2\right)\left(0.5\mathrm{m}\right)\left(\frac{1\mathrm{J}}{1\mathrm{kg}.{\mathrm{m}}^2/{\mathrm{s}}^2}\right) \\ =490\mathrm{J} \end{gathered}$$

The net change in the internal energy is,

$∆U=∆U_2-U_1$

Substitute 490J for $U_2$, and 2844.82 J for$U_1$ in the above equation.

$$\begin{gathered} ∆U=490\mathrm{J}-2844.82\mathrm{J} \\ =-2354.82\mathrm{J} \end{gathered}$$

Thus, the net change in internal energy for Tarzan from just before his feet touch the ground to when he is in the crouched position is$-2354.82\mathrm{J}$.