91Ó°ÊÓ

The snowboarder moves down the slope, which is at an angle of \(37.1^{\circ}\) with the horizontal. Therefore, we have to consider the components of the gravitational force acting along the direction parallel to the slope and perpendicular to the slope. The gravitational force can be written as \(F_g = mg\), where \(m\) is the mass of the snowboarder and \(g\) is the acceleration due to gravity. The component of gravitational force acting parallel to the slope can be found using the formula \(F_{g\parallel} = mg\sin(\theta)\), and the component perpendicular to the slope can be found using the formula \(F_{g\perp} = mg\cos(\theta)\). The normal force exerted by the slope on the snowboarder acts perpendicular to the slope, in the opposite direction of the \(F_{g\perp}\) component of gravitational force. Therefore, the normal force can be written as \(F_n = F_{g\perp}\). The frictional force acting on the snowboarder can be found using the formula \(F_f = \mu F_n\), where \(\mu\) is the coefficient of kinetic friction.

The work done by a force is given by the formula \(W = Fd\cos(\alpha)\), where \(F\) is the force applied, \(d\) is the displacement, and \(\alpha\) is the angle between the force and the displacement. For the gravitational force acting parallel to the slope, \(\alpha = 0^{\circ}\), so the work done by this force is \(W_{g\parallel} = F_{g\parallel} d\cos(0^{\circ}) = F_{g\parallel} d\). For the frictional force, the angle between the force and displacement (which is along the slope) is \(180^{\circ}\), so the work done by this force is \(W_f = F_fd\cos(180^{\circ}) = -F_fd\). The normal force acts perpendicular to the displacement, so the work done by this force is zero, \(W_n = 0\). Now we need to calculate the total work done on the snowboarder, which is the sum of the work done by each of these forces: \(W_{total} = W_{g\parallel} + W_f + W_n\).

To find the displacement, we need to know the distance the snowboarder traveled down the slope in 5.72 seconds. We can find this by using the equation \(d = v_0 t + \frac{1}{2}at^2\), where \(v_0\) is the initial speed, \(t\) is the time, and \(a\) is the acceleration. The acceleration can be found using the equation \(a = \frac{F_{net}}{m}\), where \(F_{net}\) is the net force acting on the snowboarder and \(m\) is its mass. The net force can be calculated as the difference between the gravitational force parallel to the slope (\(F_{g\parallel}\)) and the frictional force (\(F_f\)): \(F_{net} = F_{g\parallel} - F_f\). Then, we can find the displacement and the work done by each force and calculate the total work done.

Using the work-energy theorem, the change in kinetic energy is equal to the total work done on the snowboarder: \(\Delta KE = W_{total}\). The final answer will be the net work done on the snowboarder during the first 5.72 seconds of the descent.