An 82.0 kg skydiver jumps out of a balloon at an altitude of 1 500 m and opens the parachute at an altitude of 195 m.
(a) Assuming that the total retarding force on the diver is constant at 55.0 N with the parachute closed and constant at 3 600 N with the parachute open, what is the speed of the diver when he lands on the ground?
(B) At what height should the parachute be opened so that the final speed of the skydiver when he hits the ground is 5.00 m/s?
To solve these problems, we can use the principles of physics related to motion, particularly the equations of motion and the concept of work done by force.
(a) To find the speed of the diver when he lands on the ground, we need to calculate the change in kinetic energy. This can be done by considering the work done by the retarding forces during two stages: when the parachute is closed and when it is open.
When the parachute is closed, the work done is equal to the force (55.0 N) multiplied by the distance traveled (1,500 m - 195 m = 1,305 m):
Work_done = force * distance = 55.0 N * 1,305 m
This work is equal to the change in kinetic energy:
Change_in_kinetic_energy = Work_done
The change in kinetic energy is also equal to the final kinetic energy minus the initial kinetic energy:
Change_in_kinetic_energy = Final_kinetic_energy - Initial_kinetic_energy
When the parachute is closed, the initial kinetic energy is zero because the diver starts from rest. Therefore, we can rewrite the equation as:
Final_kinetic_energy = Change_in_kinetic_energy + Initial_kinetic_energy = Change_in_kinetic_energy + 0
Using the work done and the assumed constant force, we can find the final kinetic energy:
Final_kinetic_energy = 55.0 N * 1,305 m
Now we can use the equation for kinetic energy to find the final speed:
Final_kinetic_energy = (1/2) * mass * final_speed^2
Simplifying the equation, we have:
(1/2) * mass * final_speed^2 = 55.0 N * 1,305 m
Rearranging the equation to find the final speed:
final_speed^2 = (55.0 N * 1,305 m) / (1/2 * mass)
final_speed = square_root((55.0 N * 1,305 m) / (1/2 * mass))
Substituting the given values for mass:
mass = 82.0 kg
final_speed = square_root((55.0 N * 1,305 m) / (1/2 * 82.0 kg))
Calculating the final speed will give us the answer for part (a).
(b) To determine the height at which the parachute should be opened, we need to consider the work done again, but this time we will use the final speed of 5.00 m/s.
Using the equation for kinetic energy:
Final_kinetic_energy = (1/2) * mass * final_speed^2
(1/2) * mass * final_speed^2 = 3600 N * (1500 m - height)
Simplifying the equation, we have:
final_speed^2 = 7200 N * (1500 m - height) / mass
The height at which the parachute should be opened is when the final speed is 5.00 m/s. So we can substitute the values into the equation as follows:
5.00 m/s^2 = 7200 N * (1500 m - height) / mass
Rearranging the equation to solve for the height:
height = 1500 m - (mass * final_speed^2) / (7200 N)
Substituting the given values for mass and final speed, we can calculate the height at which the parachute should be opened.
By performing these calculations, we can find the answers to both parts (a) and (b) of the problem.