We're not there yet. If the spring stretches 30 m, the mechanical energy stored in the spring is ##{1\over 2}\,k\,x^2 = {1\over 2} \, 100 \, {\rm kg}/{\rm s}^2 \, \left ( 30 {\rm m} \right )^2##, which is not, repeat NOT 1500 kg m
2/s
2.
There is only ONE force acting on the spring: the pulling force that the person (in the original post it is said that it is a man) executes on the spring. There are TWO forces acting on the man.
The forces are in opposite direction as the string is stretched down, because the restoring force is pointing back up and the gravitational force is pointing down. However, the restoring force is stronger than the force of gravity. Therefore, we get an upward acceleration of 20m/s^2.
Is correct, if you are talking about the forces on the man at the bottom point (and if you use g = 10 m/s
2).
There is a difference between the kinetic energy of the spring, and total kinetic energy of the system.
Yes, there sure is. In fact the total kinetic energy in the system is the total kinetic energy of the man. (The cord is considered massless). The kinetic energy starts with zero and starts to increase when he jumps.
It keeps increasing even if the distance jumped is equal to the length of the spring/coil. At that point the acceleration is still g, so considerable. That means the speed keeps increasing, hence the kinetic energy too.
It is important to understand that the speed (and hence the kinetic energy)
keeps increasing until the
acceleration of the man is zero. That is (because F = m a) at the point where the force, exercised on the man by the stretched cord is equal in magnitude to the force, exercised (in the opposite direction) on the man by gravity. In fact, we can calculate how far the spring is stretched at that point. And we have to do that, because we will need that amount of stretch later on. Oblige me and do this calculation.
However, in certain situations, the total kinetic energy can equal to just the spring when the only force that is acting on the spring is the restoring force of the spring.
This is definitely NOT correct. I can't even understand what you try to express here.
The coil exercises a force k x on whatever is pulling on it
at all times when the stretch x ##\ge## 0.
Yet, In our scenario, we have two forces acting on the spring. If we knew the height of the bridge and the height of the string before it stretched we could find the Kinetic Energy there. (That is where Force restore -Force of gravity = ma(zero).
Definitely not true. At that point force from the spring is zero.
As the lady falls right before the spring stretches that is also where our other highest velocity is at
No, as explained above. And is was a man when he jumped
But as the string starts to stretch the velocity goes to zero
Yes! It will in fact even change sign at the bottom point and continue to increase (if you take upwards as the positive direction).
We can have two kinetic energies
No we can not. There is only one object with mass in the entire exercise: the man.
I don't have time to comment on all that.
Maybe somewhere in between
You are absolutely right ! Are you now ready to abandon the idea that the maximum upward speed is reached at the point where the cord is no longer stretched ? Because from the point where the speed
is maximum to the point where k x = 0 , the force on the man from gravity has been bigger in magnitude than the upward pulling force from the cord, so the acceleration has been downward, meaning the upward speed has been going down!
Are you familiar with acceleration a is the time derivative of velocity v : ##\ \vec a = {d\vec v\over dt}## ?
Do you know how to find extrema of functions x(y) by requiring ##{dy\over dx}=0## ?
Do you see a similarity with your exercise ?
