The concepts in this problem are similar to those in Multiple-Concept Example 4, except that the force doing the work in this problem is the tension in the cable. A rescue helicopter lifts a 85.8-kg person straight up by means of a cable. The person has an upward acceleration of 0.643 m/s2 and is lifted from rest through a distance of 13.1 m. (a) What is the tension in the cable? How much work is done by (b) the tension in the cable and (c) the person's weight? (d) Use the work-energy theorem and find the final speed of the person.

To solve this problem, we can use the principles of work, energy, and Newton's second law.

(a) The tension in the cable can be found using Newton's second law: ∑F = ma, where ∑F is the net force, m is the mass, and a is the acceleration. In this case, the net force is equal to the tension in the cable (since it is the only force acting vertically), the mass is 85.8 kg, and the acceleration is 0.643 m/s^2. Therefore, we have Tension = mass × acceleration = 85.8 kg × 0.643 m/s^2.

(b) To find the work done by the tension in the cable, we can use the equation W = F × d × cos(θ), where W is the work done, F is the force applied, d is the displacement, and θ is the angle between the force and displacement vectors. In this case, the angle between the force and displacement vectors is 0 degrees, since the force and displacement are both vertical. Since the force is the tension in the cable (which we found in part (a)), and the displacement is 13.1 m, we can calculate the work done as Work = Tension × displacement × cos(0).

(c) The work done by the person's weight can be calculated in a similar way as part (b). The force due to the person's weight is their mass times the acceleration due to gravity, which is approximately 9.8 m/s^2. Therefore, the work done by the person's weight is Weight × displacement × cos(θ), where Weight = mass × gravity = 85.8 kg × 9.8 m/s^2.

(d) Finally, we can use the work-energy theorem to find the final speed of the person. The work-energy theorem states that the work done on an object is equal to its change in kinetic energy. In this case, the work done by the tension in the cable (found in part (b)) will be equal to the change in the person's kinetic energy. We can use the equation W = ΔKE = 0.5 × m × (vf^2 - vi^2), where ΔKE is the change in kinetic energy, m is the person's mass, vf is the final velocity, and vi is the initial velocity (which is 0 m/s since the person starts from rest). Solving for vf will give us the final speed of the person.

By following these steps, you should be able to find the tension in the cable, the work done by the tension, the work done by the person's weight, and the final speed of the person in the given scenario.