Finding Max Spring Compression for Firefighter Fall

[Daemonw2ngs]

Homework Statement

A 60.kg firefighter slides down a pole while a constant frictional force of 300N retards his motion. A horizontal 20.0kg platform is supported by a spring at the bottom of the pole to cushion the fall. The firefghter stars from rest 5.00m above the platform, and the spring constant is 2500N/m. Find the maxiumum distance the spring is compressed.


2. Attempt at solution
I did this multiple times, and hte method that made most sense to me was to find difference in work, energy after loss, and the speed just before hitting the platform, THEN to use momentum to calculate the speed of both the person and the board. Using that, find the max compression of the spring.. There are 5 choices of answers

a. 0.25m
b. 2.15m
c. 1.14m
d. 47 m
e. 105m

I'm pretty sure I've doen something wrong. Any help will be gladly appreciated

 

[hage567]

You can use conservation of energy for this problem. I don’t think you need to worry about momentum or the velocity of impact. You can find the potential energy of the firefighter with respect to the platform, and you can easily work out what he loses to friction as he goes down the pole. The result of this is the energy that can be used to compress the spring, so you can find out the distance it moves. I think you also have to consider how much the weight of the platform itself is compressing the spring, before the firefighter lands on it. The sum of these two components will be the total compression of the spring.

 

[m2003]

Assuming there are no external forces acting on the system, we can use the conservation of energy principle to solve this problem. Initially, the firefighter has potential energy due to his height above the platform, which is converted into kinetic energy as he slides down the pole. However, the frictional force acting against him reduces the amount of kinetic energy he has when he reaches the platform.

To find the maximum compression of the spring, we need to find the point where all of the firefighter's kinetic energy is transferred into the spring's potential energy. This occurs when his velocity is reduced to zero, meaning all of his kinetic energy has been converted into potential energy of the spring.

Using the conservation of energy equation, we can set the initial potential energy equal to the final potential energy of the spring:

mgh = 1/2kx^2

Where m is the mass of the firefighter, g is the acceleration due to gravity, h is the initial height of the firefighter, k is the spring constant, and x is the maximum compression distance of the spring.

Substituting in the values given in the problem, we get:

(60kg)(9.8m/s^2)(5.00m) = 1/2(2500N/m)(x^2)

2940 = 1250x^2

x^2 = 2.352

x = √2.352 = 1.53m

Therefore, the maximum compression distance of the spring is 1.53m, which is closest to answer choice C.