Chopstick

Simplifications: I think you can ignore friction at first, since you're talking about nice rolling things. A spring on the end of a rod is easier to model than bungee, since that confines stretching to approximately a single point. The starting state is with the spring at its' resting point, everything else is rigid. And the ending state, after the acceleration is done, is a steady state where bike & car have the same velocity, with 0 acceleration. Oh, also assume the car has uniform acceleration. For the math, I'm using arbitray time dimensions (ticks; p[resumably significantly less than 1 second per tick), but that should be okay up to a scaling factor. And PS, car is much more massive than bike, bike is much more massive than spring Exact answer does require chemical or material knowledge to determine k Since you want to know about the dynamic situation between the steady states, I would assume a numerical/simulation in discrete time is necessary. Tick 0: everything has v=0, displacement of spring is 0, the car has acceleration a, everything else has acceleration 0. tick 1: car has velocity v, car has traveled distance v/2. v=at, x=0.5 at^2. The string is now stretched to length y = 0 + x. F=kx gives you the force on the spring. Assuming no kind of damping factor, the string-stretching force applies to spring & bike. F=m'a' gives you an acceleration for the bike (primes are values for the bike, unprimed is car) tick 2: car has velocity 2v, bike has velocity v', car is at position x=2a, bike is position x'=0.5 a'. Spring length is X - X'. Calculate force on spring, repeat calculation for spring force F, bike accelerationj and velocity a' and v' repeat calculations every tick, untiil car velocity and bike velocity are however close you want them to be. That's my rough draft of a procedure at least.