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
The tension is mg+ma
The work done by tension is tension*13
work done by weigh? mg*13
worktension=changePE+KEchange
solve for KE change, then velocity from that.
How do you solve for KE change?
Work done by weight- 65*9.8*13= 8281 right?
To find the answers to the given questions, we can use the concepts of force, work, and energy.
(a) To find the tension in the cable, we can use Newton's second law of motion, which states that the net force acting on an object equals the mass of the object multiplied by its acceleration. In this case, the net force is equal to the tension in the cable, and the mass of the person is 65 kg. The person's upward acceleration is given as 0.70 m/s^2. Therefore, the tension in the cable can be calculated as follows:
Tension = mass * acceleration
Tension = 65 kg * 0.70 m/s^2
Tension = 45.5 N
Therefore, the tension in the cable is 45.5 Newtons.
(b) To find the work done by the tension in the cable, we can use the formula:
Work = force * distance.
In this case, the force is the tension in the cable, which we found to be 45.5 N. The distance is given as 13 m. Plugging in the values, we get:
Work = 45.5 N * 13 m
Work = 591.5 J
Therefore, the work done by the tension in the cable is 591.5 Joules.
(c) To find the work done by the person's weight, we need to calculate the gravitational potential energy. The work done is equal to the change in potential energy. The potential energy is given by the formula:
Potential Energy = mass * gravity * height.
The mass of the person is 65 kg, the acceleration due to gravity is approximately 9.8 m/s^2, and the height is given as 13 m.
Potential Energy = 65 kg * 9.8 m/s^2 * 13 m
Potential Energy = 8,290 J
Therefore, the work done by the person's weight (gravity) is 8,290 Joules.
(d) According to the work-energy theorem, the work done on an object is equal to the change in its kinetic energy. Since the person was initially at rest, the initial kinetic energy is 0. The work done on the person is equal to the sum of the work done by the tension in the cable and the work done by the person's weight. So, the total work done is:
Total Work = Work done by tension + Work done by gravity
Total Work = 591.5 J + 8,290 J
Total Work = 8,881.5 J
According to the work-energy theorem, the total work done is equal to the change in kinetic energy. Therefore:
Change in Kinetic Energy = Total Work
(1/2) * mass * final velocity^2 = 8,881.5 J
We know that the mass of the person is 65 kg. Rearranging the equation, we have:
final velocity^2 = (2 * total work) / mass
final velocity^2 = (2 * 8,881.5 J) / 65 kg
final velocity^2 = 273.1 m^2/s^2
Taking the square root of both sides, we get:
final velocity = sqrt(273.1 m^2/s^2)
final velocity ≈ 16.52 m/s
Therefore, the final speed of the person is approximately 16.52 m/s.