Using an electric winch to compress a spring and launch an object I'm having trouble grokking the relationship between a winch's pull/torque and a spring's potential energy.
I would like to compress a spring using an electric winch and figure out how far it will be able to launch an object.


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*If the spring's maximum load is higher than the winch's force, do the other details of the spring even matter? Specifically, if the electric winch has 2000-lbs of pull, is the maximum potential energy it can store in the spring 2000-lbs (8896-N) of force, regardless of the spring's length or constant?

*What is the correct formula for finding the height which the spring will launch an object (of let's say 1-pound) straight into the air (after it has been compressed up to the winch's max pull of 2000-lbs)?
 A: *

*I am assuming that by 'winch's force' you mean the maximum force that can be generated by the torque produced in the winch. If this force is not higher than the load, then no it cannot pull it. The maximum potential energy the winch can hold depends on how much turns it is given, the properties of the spring are totally irrelevant here.

*w = mgh where m is the mass of the object, h is the maximum height and w is the work done to completely compress the spring.
A: The energy stored in the spring is equal to the work done in compressing it.  The force needed to compress the spring to the maximum is only a small part of this calculation.
Suppose the spring is completely relaxed, and you apply a small force compressing the spring slightly.  The energy stored is the product of the force and the distance, measured in Newton-metres, or joules.
Now you increase the force slightly, and the spring compresses a short distance more.  The additional energy stored in this new force multiplied by this next compression distance.
This continues, until the spring is totally compressed.  So to calculate the energy stored, you need to know how far the spring compresses, and how the compression force changes as the spring compresses.
Edit to rely to comment:
Suppose the spring is quite long in its relaxed state, with a spring constant of 40 lbf/in.  The spring will need to compress by 50 in, to reach the winch maximum of 2000 lbf..  So the energy stored in the compressed spring would be (with suitable conversion to SI) $$E_s=\frac12kx^2=\frac127005\times1.27^2=5650 \text{ joules}$$
So this would be the energy available to loft a projectile.  The problem would be to transfer as much as possible of this energy to a light projectile.  There are inherent, internal losses in the rapid extension of a spring that make it difficult to efficiently launch a missile.  Usually, the spring pushes slowly, with huge force on the short end of a lever, while a light mass accelerates quickly at the long end of the lever.
Google "trebuchet", or even "punkin chuckin".  Spelling is correct!
The equation in the comment would give the maximum height possible, with no losses in the spring.
