Mechanical vibrations

A mass of $2$ kilograms is on a spring with spring constant $k$ newtons per meter with no damping. Suppose the system is at rest and at time $t=0$ the mass is kicked and starts traveling at 2 meters per second. How large does $k$ have to be so that the mass does not go further than 3 meters from the rest position?

Suppose we have an RLC circuit with a resistor of 100 milliohms (0.1 ohms), inductor of inductance of 50 millihenries (0.05 henries), and a capacitor of 5 farads, with constant voltage.

1. Set up the ODE equation for the current $I$.
2. Find the general solution.
3. Solve for $I(0) = 10$ and $I'(0) = 0$.

A 5000 kg railcar hits a bumper (a spring) at 1 m/s, and the spring compresses by 0.1 m. Assume no damping.

1. Find $k$.
2. How far does the spring compress when a 10000 kg railcar hits the spring at the same speed?
3. If the spring would break if it compresses further than 0.3 m, what is the maximum mass of a railcar that can hit it at 1 m/s?
4. What is the maximum mass of a railcar that can hit the spring withou…

A mass of $m$ kg is on a spring with $k=3 N/m$ and $c=2 Ns/m$. Find the mass $m_0$ for which there is critical damping. If $m < m_0$, does the system oscillate or not, that is, is it underdamped or overdamped?