91Ó°ÊÓ

The mass of the bear is 25 kg.

The distance travelled by bear is 12 m.

The speed of the bear just before hitting the ground is 5.6 m/s.

Total energy of the bear – earth system is conserved. The potential energy at the top of the tree is converted to kinetic energy as it reaches the bottom, and some amount of energy is lost during the motion due to friction.

Formulae are as follow:

Potential energy,$PE=mgh$

Kinetic energy,$KE=\frac{1}{2}m{v}^2$

$\mathrm{Frictional}\mathrm{force}=\mathrm{Energy}\mathrm{lost}/\mathrm{distance}\mathrm{travelled}$

The potential energy of the bear at the top of the tree is given by,

$$\begin{gathered} PE=U_f-U_i \\ =-mgh \\ =-25\mathrm{kg}×9.8\mathrm{m}/{\mathrm{s}}^2×12\mathrm{m} \\ =-2.9×{10}^3\mathrm{J} \\ =-2.9\mathrm{kJ} \end{gathered}$$

Hence, the change in gravitational energy of the bear-earth system is -2.9 kJ.

Thekinetic energy of the bear just before hitting the ground is given by,

$$\begin{gathered} KE=\frac{1}{2}m{v}^2 \\ =\frac{1}{2}×25\mathrm{kg}×{\left(5.6\mathrm{m}/\mathrm{s}\right)}^2 \\ =3.9×{10}^2\mathrm{J} \end{gathered}$$

Hence, the kinetic energy of the bear just before hitting the ground is $3.9×{10}^2\mathrm{J}$.

Due to frictional force acting on the bear, some of its initial PE is lost during its travel to the bottom of the tree. Thus, the energy lost during the travel is calculated as,

$$\begin{gathered} ∆E=PE\left(\mathrm{at}\mathrm{the}\mathrm{top}\right)-KE\left(\mathrm{at}\mathrm{the}\mathrm{bottom}\right) \\ ∆\mathrm{E}=2.9×{10}^3-3.9×{10}^2 \\ =2510\mathrm{J} \end{gathered}$$

This energy loss is related to the frictional force as,

$$\begin{gathered} ∆E=F_{f}d∆F_f \\ =\frac{∆E}{d} \\ =\frac{2510}{12} \\ =2.10×{10}^2\mathrm{N} \end{gathered}$$

Hence, the energy loss is related to the frictional forceis$2.10×{10}^2\mathrm{N}$.

Therefore, the law of conservation of energy holds true for the bear-earth system. The total energy at the top of the tree is potential energy, some part of which is converted to kinetic energy at the bottom of the tree, and some part is lost due to the frictional force acting on the bear.