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

To solve this problem, we can use the work-energy theorem, which states that the work done on an object is equal to the change in its kinetic energy.

Step 1: Find the force exerted by the cable (tension).
Using Newton's second law, we can find the force exerted by the cable.
Force = mass * acceleration
Force = 65 kg * 0.70 m/s^2
Force = 45.5 N

Step 2: Calculate the work done by the tension in the cable.
The work done is equal to the force applied multiplied by the displacement.
Work = Force * displacement
Work = 45.5 N * 13 m
Work = 591.5 J

Step 3: Find the work done by the person's weight.
The work done by a constant force (such as gravity) can be calculated as the product of the force, displacement, and the cosine of the angle between them. In this case, the angle is 0 degrees (straight up).
Work = Force * displacement * cos(angle)
Work = weight * displacement * cos(0)
Work = 65 kg * 9.8 m/s^2 * 13 m

Step 4: Use the work-energy theorem to find the final speed of the person.
The total work done on the person is equal to the change in kinetic energy.
Total work = Work done by tension + Work done by weight
Total work = 591.5 J + (65 kg * 9.8 m/s^2 * 13 m)

Since the work done by tension is equal to the total work, the net work done is zero. Therefore, the change in kinetic energy is zero.
Change in kinetic energy = 0
Final kinetic energy - Initial kinetic energy = 0
(1/2) * mass * final velocity^2 - (1/2) * mass * initial velocity^2 = 0

Since the person starts from rest (initial velocity = 0), the equation simplifies to:
(1/2) * mass * final velocity^2 = 0

Solving for final velocity:
final velocity = sqrt((2 * 0) / 65 kg)
final velocity = 0 m/s (since the person reaches a stop at the top)

Therefore, the final speed of the person is 0 m/s.

To solve this problem, we can use the work-energy theorem, which states that the work done on an object is equal to its change in kinetic energy.

(a) The tension in the cable can be found using Newton's second law, which states that the net force on an object is equal to its mass multiplied by its acceleration. In this case, the net force is equal to the tension in the cable, so we can write:
Tension = mass * acceleration
Tension = 65 kg * 0.70 m/s^2
Tension = 45.5 N

Therefore, the tension in the cable is 45.5 N.

(b) The work done by the tension in the cable can be calculated using the formula:
Work = force * distance * cos(theta)
In this case, the force is the tension in the cable, the distance is the height the person is lifted (13 m), and the angle between the force and the direction of displacement is 0 degrees since the force is acting in the same direction as the displacement:
Work = tension * distance * cos(0)
Work = 45.5 N * 13 m * cos(0)
Work = 591.5 J

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

(c) The work done by the person's weight can be calculated using the formula:
Work = force * distance * cos(theta)
In this case, the force is the person's weight, the distance is the height the person is lifted (13 m), and the angle between the force and the direction of displacement is 180 degrees since the force is acting in the opposite direction as the displacement:
Work = weight * distance * cos(180)
Work = mass * gravity * distance * cos(180)
Work = 65 kg * 9.8 m/s^2 * 13 m * cos(180)
Work = -12674 J

Since the angle between the force and the displacement is 180 degrees, the work done by the person's weight is negative (-12674 J).

(d) According to the work-energy theorem, the work done on an object is equal to its change in kinetic energy. Therefore, we can write:
Work done by tension + Work done by weight = Change in kinetic energy

Since the person starts from rest and is lifted straight up, the change in kinetic energy is equal to the final kinetic energy. We can assume the initial kinetic energy is zero.

The work done by tension is 591.5 J and the work done by weight is -12674 J, so we can write:
591.5 J - 12674 J = 0.5 * mass * final velocity^2

Simplifying the equation, we get:
0.5 * 65 kg * final velocity^2 = -12182.5 J

Solving for final velocity, we have:
final velocity^2 = (-12182.5 J) / (0.5 * 65 kg)
final velocity^2 = -374.94 m^2/s^2

Since velocity cannot be negative, the final velocity is zero.

Therefore, the final speed of the person is 0 m/s.