91Ó°ÊÓ

To calculate the initial kinetic energy (KE) of the skier, we can use the formula: \[KE = \frac{1}{2} mv^2\] where m is the mass of the skier (60 kg) and v is the initial speed of the skier (12 m/s). Plugging in the values, we get: \[KE = \frac{1}{2} (60 \mathrm{kg}) (12\,\mathrm{m/s})^2 = 4320\, \mathrm{J}\]

To calculate the potential energy (PE) gained when the skier reaches the top of the hill, we can use the formula: \[PE = mgh\] where m is the mass of the skier (60 kg), g is the gravitational acceleration (approximately 9.81 m/s²), and h is the height of the hill (2.5 m). Plugging in the values, we get: \[PE = (60\, \mathrm{kg})(9.81\,\mathrm{m/s^2})(2.5\, \mathrm{m})= 1471.5\, \mathrm{J}\]

To calculate the work done against friction, use the formula: \[W_{friction} = F_{friction} \times d\] The force of friction can be calculated as: \[F_{friction} = \mu F_N\] where \(F_N\) is the normal force, which is equal to the weight of the skier (mg) and \(\mu\) is the coefficient of friction (0.80). First, we should find the distance that the skier traveled, which can be done using the Pythagorean theorem: \[d = \sqrt{(\Delta x)^2 + (\Delta h)^2}\] Assuming the hill is inclined at an angle of θ, the work done against friction can be calculated as: \[W_{friction} = F_{friction} \times d \times \cos(\theta)\] However, we can simplify this calculation by noting that the force of friction is opposite to the direction of motion and proportional to the normal force (which is proportional to the weight of the skier). So, we can write: \[W_{friction} = \mu mgh\] Plugging in the values, we get: \[W_{friction} = (0.80)(60\, \mathrm{kg})(9.81\,\mathrm{m/s^2})(2.5\, \mathrm{m})= 1177.2\, \mathrm{J}\]

According to the conservation of energy principle, the initial kinetic energy (KE) should equal the sum of the potential energy gained (PE) and the work done against friction (W_friction). Therefore, the final kinetic energy (KE_final) can be found using the equation: \[KE_{final} = KE - PE - W_{friction}\] Plugging in the values, we get: \[KE_{final} = 4320\,\mathrm{J} - 1471.5\,\mathrm{J} - 1177.2\,\mathrm{J}= 1671.3\, \mathrm{J}\] Now, we can find the final speed (v_final) using the relationship between kinetic energy and speed: \[KE_{final} = \frac{1}{2}mv_{final}^2\] Solving for v_final, we get: \[v_{final} = \sqrt{\frac{2KE_{final}}{m}}\] Plugging in the values, we get: \[v_{final}= \sqrt{\frac{2(1671.3\,\mathrm{J})}{(60\,\mathrm{kg})}} = 7.53\,\mathrm{m/s}\] So, the skier's final speed at the top of the hill is approximately 7.53 m/s.