Final answer:
To find the speed of a 62.5 kg rider at the bottom of the slide, conservation of energy is used while accounting for potential energy from spring compression, gravitational potential energy, and the work done by friction. Several calculations involving spring energy, gravitational energy, and friction must be combined to solve for the final kinetic energy, from which the rider's speed at the bottom can be determined.
Step-by-step explanation:
To determine the speed of a 62.5 kg rider at the bottom of the water slide, we will need to apply the principles of conservation of energy and consider the work done by nonconservative forces like friction. The potential energy (PE) at the top due to the spring will be converted into kinetic energy (KE) and potential energy due to height, and some of this energy will be used to do work against friction.
The spring potential energy (PEspring) can be calculated by the formula PEspring = ½ k x2, where k is the spring constant and x is the compression distance. Given k = 2.00E+03 N/m and x = 2.10 m, we find:
PEspring = ½ (2.00E+03 N/m)(2.10 m)2 = 4.41E+03 J
The gravitational potential energy (PEgrav) at the top of the slide is given by PEgrav = mgh, where m is the mass, g is the acceleration due to gravity (9.81 m/s2), and h is the height. With m = 62.5 kg and h= 49.0 m, we have:
PEgrav = (62.5 kg)(9.81 m/s2)(49.0 m) = 3.02E+04 J
The work done by friction (Wfriction) is Wfriction = d, where is the kinetic friction force and d is the distance traveled. The force of kinetic friction can be found from = μk N, and the normal force (N) on a slope is N = m g cos(θ). With μk = 0.0500 and θ = 58 degrees, we must also calculate d using trigonometry.
Since we have no initial velocity and no non-conservative work other than friction, the final kinetic energy (KEfinal) at the bottom will be equal to the total initial mechanical energy (PEspring + PEgrav) minus the work done by friction (Wfriction). To find the final speed (v), we use KEfinal = ½ mv2.
We must perform several calculations, taking trigonometry and friction into account, to arrive at the final speed.