Solving Spring/Work Problem: Find Min. Work Required

[Tabe]

The exact problem is:
A massless spring is between a 1-kilogram mass and a 3-kilogram mass, but it is not attached to either mass. Both masses are on a horizontal frictionless table. In an experiment, the 1-kilogram mass is held in place, and the spring is compressed by pushing on the 3-kilogram mass. The 3-kilogram mass is then released and moves off with a speed of 10 meters per second.

(a) Determine the minimum work needed to compress the spring in this experiment.

I am totally lost on this problem. I mean, I understand what it is asking for, but I don't know which equation to use and what variables to solve for. I know the equation for work is W= 1/2kx^2 but how do I solve for either variable?
I was looking through my notes and saw several formulas that I could use, but each one is missing the same varible, either force or the distance.

Can someone please point me in the right direction...

 

[Pyrrhus]

Consider conservation of Mechanical energy, when the spring is compressed with mass of 3 kg, then there's only spring potential energy, when the mass is released with a speed of 10m/s then there's only kinetic energy. Use both initial and final case, to find the Work that should be done on the spring or the change in spring potential energy.

 

[quicknote]

I would suggest approaching this problem by breaking it down into smaller, more manageable steps. First, let's identify the known variables in the problem: the masses (1 kg and 3 kg), the speed (10 m/s), and the fact that the spring is massless and on a frictionless table.

Next, let's consider the concept of work. Work is defined as the force applied over a certain distance. In this case, the force is being applied by the 3 kg mass as it compresses the spring, and the distance is the amount the spring is compressed. Therefore, we can use the equation W = Fd to solve for the work needed to compress the spring.

To find the force, we can use Newton's second law, F=ma, where m is the mass of the 3 kg object and a is the acceleration. Since the object is initially at rest and then moves with a speed of 10 m/s, we can use the equation v^2 = u^2 + 2as to solve for the acceleration (where u is the initial velocity, which is 0 m/s).

Once we have the force, we can plug it into the work equation along with the distance that the spring is compressed (which we can calculate using the given equation for the spring, W=1/2kx^2). This will give us the minimum work required to compress the spring in this experiment.

I hope this helps guide you in the right direction. Remember, when solving problems in science, it's important to break them down into smaller steps and use the appropriate equations for the known variables. Good luck!