91Ó°ÊÓ

Calculating the GPE gives us: \(GPE = (63 \, \mathrm{kg}) (9.81 \, \mathrm{m/s^2}) (26.60 \, \mathrm{m}) \approx 16461.26 \, \mathrm{J}\) (joules). #tag_title# Step 2: Determine Kinetic Energy at Bottom of Slope #tag_content# Since there is no friction on the slope, we can apply the conservation of mechanical energy and assume that the GPE at the top of the slope will be completely converted into kinetic energy (KE) at the bottom. Thus, KE at the bottom of the slope will be equal to the GPE at the top: \(KE = 16461.26 \, \mathrm{J}\). #tag_title# Step 3: Determine Speed at Bottom of Slope #tag_content# We can now calculate the skier's speed at the bottom of the slope using the kinetic energy formula: \(KE = \frac{1}{2}mv^2\), where \(m\) is the mass of the skier (63 kg) and \(v\) is the speed. Solving for \(v\), we get: \(v = \sqrt{\frac{2 \cdot KE}{m}} = \sqrt{\frac{2 \cdot 16461.26 \, \mathrm{J}}{63 \, \mathrm{kg}}} \approx 22.97 \, \mathrm{m/s}\) The skier's speed at the bottom of the slope is approximately \(22.97 \, \mathrm{m/s}\). #tag_title# Step 4: Determine Deceleration on Horizontal Surface #tag_content# To find the deceleration on the horizontal surface, we can use the formula: \(v^2 = u^2 + 2as\), where \(u\) is the initial speed (\(22.97 \, \mathrm{m/s}\)), \(v\) is the final speed (0 m/s, since the skier comes to rest), \(a\) is the deceleration, and \(s\) is the distance traveled on the horizontal surface (\(96 \, \mathrm{m}\)). Solving for \(a\), we get: \(a = \frac{v^2 - u^2}{2s} = \frac{0^2 - (22.97 \, \mathrm{m/s})^2}{2(96 \, \mathrm{m})} \approx -2.74 \, \mathrm{m/s^2}\) The deceleration on the horizontal surface is approximately -2.74 m/s² (negative sign indicates the direction opposite to the motion). #tag_title# Step 5: Determine Coefficient of Kinetic Friction #tag_content# We can now use Newton's second law of motion to find the friction force acting on the skier: \(F_f = ma\), where \(F_f\) is the friction force, \(m\) is the skier's mass (63 kg), and \(a\) is the deceleration (-2.74 m/s²). The friction force is: \(F_f = (63 \, \mathrm{kg})(-2.74 \, \mathrm{m/s^2}) = -172.62 \, \mathrm{N}\) (negative sign indicates the direction opposite to the motion). Next, we use the formula for kinetic friction: \(F_f = \mu F_n\), where \(F_n\) is the normal force (equivalent to the weight of the skier in this case) and \(\mu\) is the coefficient of kinetic friction. We have: \(-172.62 \, \mathrm{N} = \mu (mg)\) Solving for \(\mu\), we get: \(\mu = \frac{-172.62 \, \mathrm{N}}{(63 \, \mathrm{kg})(9.81 \, \mathrm{m/s^2})} \approx 0.28\) The coefficient of kinetic friction between the skier and the horizontal surface is approximately 0.28.