A cave rescue team lifts an injured spelunker directly upward and out of a sinkhole by means of a motor-driven cable. The lift is performed in three stages, each requiring a vertical distance of 7.70 m: (a) the initially stationary spelunker is accelerated to a speed of 3.80 m/s; (b) he is then lifted at the constant speed of 3.80 m/s; (c) finally he is decelerated to zero speed. How much work is done on the 62.0 kg rescue by the force lifting him during each stage?
Force on cable = m (g+a)
work done = force * distance
phase 1
average speed up v = (0+3.8)/2 =1.9 /s
t = 7.7/1.9 = 4.05 seconds
a = change in speed/change in time = 3.8/4.05 = .938 m/s^2
F = 62 (9.8+ .938) = 667 N
work = 667*7.7 = 5126 Joules
phase 2
F = m g = 62*9.8 = 608 N
F * 7.7 = 4678 Joules
phase 3
F = 62 (9.8 -.938) = 549 N
work = 549*7.7 = 4230 Joules
To calculate the work done on the rescue during each stage, you need to use the formula for work:
Work = Force * Distance * cosθ
In this case, the force is the force lifting the rescue, the distance is the vertical distance being lifted, and θ is the angle between the direction of the force and the direction of the displacement (which in this case is upward).
In stage (a), the spelunker is being accelerated. To calculate the work, you need to first find the force that is exerted on the spelunker to accelerate him. Since the spelunker is initially stationary and is accelerated to a speed of 3.80 m/s, you can use Newton's second law of motion:
Force = mass * acceleration
The mass of the rescue is given as 62.0 kg, and the acceleration can be calculated using the equation of motion:
vf^2 = vi^2 + 2 * acceleration * distance
Here, vf is the final velocity (3.80 m/s), vi is the initial velocity (0 m/s), and the distance is 7.70 m. Solve for the acceleration:
acceleration = (vf^2 - vi^2) / (2 * distance)
Substituting the given values:
acceleration = (3.80^2 - 0^2) / (2 * 7.70)
Now that you have the acceleration, you can calculate the force:
Force = mass * acceleration
Substituting the given values:
Force = 62.0 kg * acceleration
Finally, you can calculate the work done in stage (a) using the formula:
Work(a) = Force * Distance * cosθ
Since the angle θ is 0 degrees (the force is parallel to the displacement), the cosθ becomes 1:
Work(a) = Force * Distance
Substituting the values:
Work(a) = (62.0 kg * acceleration) * 7.70 m
Repeat the same process for stage (b) and stage (c) using the given values for velocity and distance, but remember that the force is constant throughout stage (b) and there is deceleration in stage (c) (so acceleration will be negative).
Work(b) = Force * Distance * cosθ
Work(c) = Force * Distance * cosθ
Once you have calculated the work done during each stage, you will have the total work done on the rescue by the force lifting him.