A rescue helicopter lifts a 65 kg person straight up by means of a cable. The person has an upward acceleration of 0.70 m/s2 and is lifted from rest through a distance of 13 m.

(a) What is the tension in the cable?
N
(b) How much work is done by the tension in the cable?
J
(c) How much work is done by the person's weight?
J
(d) Use the work-energy theorem and find the final speed of the person.
m/s

how do you solve for the final speed?

What is your thinking?

(a) Start by drawing a free body diagram for the person and applying Newton's second law, F = m a. The "F" will be the difference between the cable tension T and the weight M g. Solve for T.

(b) T*(distance moved) = work done by cable

(c) -M*g x (distance moved) = work done by person's weight

(d) Total work done = (b) + (c) = change in kinetic energy

vf= sqrt( (2 Kef)/m)

To find the answer to these questions, we can use the principles of Newton's laws of motion and work-energy theorem. Let's break it down step by step:

(a) To find the tension in the cable, we need to calculate the net force acting on the person. We can use Newton's second law, which states that the net force is equal to the mass of an object multiplied by its acceleration. In this case, the upward acceleration is 0.70 m/s^2, so the net force can be calculated as follows:

Net force = mass × acceleration
= 65 kg × 0.70 m/s^2
= 45.5 N

Since the helicopter is applying an upward force, and the mass is moving upward, the tension in the cable is equal to the net force acting on the person, which is 45.5 N.

(b) To calculate the work done by the tension in the cable, we use the formula for work: work = force × distance × cos(theta). However, in this case, the angle between the force and the direction of displacement is zero degrees (cos(0) = 1), so we can simplify the formula to: work = force × distance.

Work done by the tension = tension × distance
= 45.5 N × 13 m
= 591.5 J

Therefore, the work done by the tension in the cable is 591.5 Joules.

(c) The work done by the person's weight can be calculated using the same work formula, where the force is the weight and the distance is the height through which the person is lifted.

Work done by weight = weight × distance × cos(theta)
= m × g × distance × cos(180)
= 65 kg × 9.8 m/s^2 × 13 m × (-1)
= - 8390 J

In this case, the person is being lifted against the force of gravity, so the work done by the person's weight is negative, -8390 Joules.

(d) According to the work-energy theorem, the work done on an object is equal to the change in its kinetic energy (KE). The initial kinetic energy is zero because the person starts from rest. The work done on the person is equal to the change in kinetic energy, so we can solve for the final speed.

Work done by tension = change in kinetic energy
= (1/2) × mass × (final velocity)^2 - 0

Since the mass is 65 kg, and the initial velocity is 0, we can rearrange the formula as follows:

final velocity = square root of [2 × (work done by tension) / mass]

final velocity = square root of [2 × 591.5 J / 65 kg]
= square root of [18.15385 m^2/s^2]
= 4.26 m/s

Therefore, the final speed of the person is 4.26 m/s.