This question came up as an exercise in a first year undergraduate course I was a TA for. It turned out to be a lot more difficult (impossible?) than anticipated...
Two platforms of mass $M_1$ and $M_2$ ($M_1\neq M_2$) are connected by a spring of constant $k$, and are initially at rest with the spring unstretched, sitting on a frictionless surface. A man of mass $m$ stands on one platform and begins to run, always with a constant speed $v$ measured relative to the platform he is running on. What is the maximum speed reached by the other platform, relative to the ground?
My intuition is telling me that I need to know something about how the man gets from rest to his constant speed (instantaneously? very slowly? with some smooth acceleration?) to solve this, but I haven't been able to prove to myself that this is required. If this is a requirement, I think the most reasonable assumption would be that he reaches his full speed instantaneously.
What I'm most interested in is whether this problem can be tackled with a typical freshman's toolbox - simple arguments around conservation of energy/momentum, no/very limited differential equations, basic calculus. I can see an easy way to get an upper bound on the maximum speed from energy/momentum considerations, but I don't see a way to check if this speed is ever reached.