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 89.2-kg person straight up by means of a cable. The person has an upward acceleration of 0.702 m/s2 and is lifted from rest through a distance of 14.3 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.
(a) = 936.79N
(b) = 13395J
(c) = -12500J
but I can't figure out (d)
To find the final speed of the person, we can use the work-energy theorem. The work-energy theorem states that the work done on an object is equal to the change in its kinetic energy. In this case, the work done by the tension in the cable will be equal to the change in the person's kinetic energy.
The work done by the tension in the cable can be calculated using the formula:
Work = Force × Distance × cos(θ),
where θ is the angle between the force and the direction of motion. In this case, since the cable is being lifted straight up, θ is 0 degrees, and therefore cos(θ) = 1.
So, the work done by the tension in the cable is:
Work = Tension × Distance,
where Tension is the tension in the cable and Distance is the distance the person is lifted.
Given that the distance is 14.3 m and the tension in the cable is 936.79 N (calculated in part (a)), we can calculate the work done by the tension:
Work = 936.79 N × 14.3 m = 13395 J.
This means that the work done by the tension in the cable is 13395 J.
Now, to find the work done by the person's weight, we can use the formula:
Work = Force × Distance × cos(θ),
where in this case, the force is the person's weight and θ is the angle between the force and the direction of motion. Since the person is being lifted straight up, θ is 0 degrees, and therefore cos(θ) = 1.
The person's weight can be calculated using the formula:
Weight = mass × acceleration due to gravity,
where the mass is 89.2 kg and the acceleration due to gravity is 9.8 m/s^2.
So, the person's weight is:
Weight = 89.2 kg × 9.8 m/s^2 = 873.76 N.
Now, we can calculate the work done by the person's weight:
Work = 873.76 N × 14.3 m = 12500 J.
This means that the work done by the person's weight is -12500 J (negative because the weight is acting in the opposite direction to the motion).
Finally, to find the final speed of the person using the work-energy theorem, we can set the work done by the tension and the work done by the weight equal to the change in kinetic energy:
Work (tension) + Work (weight) = Change in Kinetic Energy.
Since there is no initial kinetic energy (the person starts from rest), the change in kinetic energy is equal to the final kinetic energy.
The final kinetic energy is given by the formula:
Kinetic Energy = (1/2) × mass × velocity^2,
where the mass is 89.2 kg.
Setting up the equation:
13395 J + (-12500 J) = (1/2) × 89.2 kg × velocity^2.
Simplifying:
895 J = 44.6 kg × velocity^2.
Now, solving for the velocity:
velocity^2 = 895 J / 44.6 kg = 20.05 m^2/s^2.
Taking the square root to find the velocity:
velocity = √(20.05 m^2/s^2) = 4.47 m/s.
Therefore, the final speed of the person is 4.47 m/s.